Math 7 – Bid Is Non Negotiable (Read First Before Bidding!!!)

 Read first before bidding.  Accept Bid as is or don’t Bid at all.  I’m not increasing so don’t ask and don’t waste my time 

Lesson 3.6

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).

 

 

· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.  

· 25. Use functional models to make predictions and solve problems. 

Specific Objectives

Students will understand that

· linear models can be created based on collected or given data.

Students will be able to

· write the equation for a linear model given a set of data.

Spring Scale

Problem Situation

1

:  Spring Scale

Watch this demonstration of hanging weights on a spring.

https://www.youtube.com/watch?time_continue=4&v=Dfrl

–

EcU0kQ

The video gave this data:

mass (grams)	50	100	150
height (cm)	25.8	21	16 .2

#1 Points possible: 5. Total attempts: 5

Using the dot tool, plot all three points from the video on the graph below.  Then, switch to the line tool and draw a line that passes through the points.  Mass is on the horizontal axis and height is on the vertical axis.

Clear All Draw: 

#2 Points possible: 5. Total attempts: 5

Write an equation for the line giving the height, h, when the mass of the weights added to the spring is w.

   

#3 Points possible: 15. Total attempts: 5

The equation tells us that when no weight is on the hanger, the height will be        , and that as weight is added, the height     by       

#4 Points possible: 5. Total attempts: 5

The mystery block of wood pulled down the hanger to

24

.5 cm. Use this with your linear model to estimate the mass of the block, rounded to 1 decimal place.

 grams

Sinking Wood

Problem Situation 2: Sinking a Block of Wood

A block of wood, 10cm by 20 cm by 4 cm thick is floated on a pool of water, submerging it to a depth of 1 cm.  A 50 gram mass is placed on top of the block, increasing the depth submerged to 1.25.  Another 50 gram mass is placed on top, increasing the depth to 1.5.

#5 Points possible: 5. Total attempts: 5

Find a linear model for the depth of the block in terms of the mass placed on top, with D representing the depth in centimeters to which the block is submerged, and w representing the amount of weight in grams on top of the block.

   

#6 Points possible: 5. Total attempts: 5

Use the model to predict the mass of the block of wood. (Measure the weight in grams.)

    grams

#

7

Points possible: 5. Total attempts: 5

How much mass could be added before the block would be totally submerged?

    grams

HW 3.6

#1 Points possible: 5. Total attempts: 5

A linear model passes through the points (20, 607) and (45, 1182).  

Find the equation of the line, with x as the input and y as the output.

   
#2 Points possible: 5. Total attempts: 5

16

t		16	24	36	40
y	1	7	19

Given the table of values above, find a linear equation for y in terms of t.

   

#3 Points possible: 16. Total attempts: 5

The population of wolves in a sanctuary has been growing by about 20 wolves a year.  In 2012, there were 415 wolves in the sanctuary.  

a. Create a linear equation for the number of wolves, P, using the input variable n = number of years since 2012.
    

b. Create a linear equation for the number of wolves, P, using the input variable t = number of years since 2010.
    

c. Predict the number of wolves in the sanctuary in 2018.  Try using both equations to ensure they agree.
 wolves

d. What are the advantages to each way of defining the input variable?

#4 Points possible: 10. Total attempts: 5

It is possible to approximate the outside temperature based on how fast crickets are chirping.  Suppose on a 50 degree day, you measure 52 chirps in a minute, and on a 74 degree day, you measure 148 chirps in a minute. 

a. Based on this data, determine a linear equation that will output the number of chirps in a minute, n, given the temperature in degrees, t.
    

b. Use your equation to predict the temperature if you heard 96 chirps in a minute, to the nearest degree.
 degrees

#5 Points possible: 10. Total attempts: 5

Match each linear equation with its graph
GKRBP

Equation

·  y=−3xy=-3x

·  y=3y=3

·  y=x+2y=x+2

·  y=−14xy=-14x

·  y=3x+2y=3x+2

Graph Color

a. blue (B)

b. black (K)

c. green (G)

d. red (R)

e. purple (P)

 

#6 Points possible: 12. Total attempts: 5

RGBK

If all the graphs above have equations with form y=mx+by=mx+b,

a. Which graph has the largest value for b?      

b. Which graph has the smallest value for b?      

c. Which graph has the largest value for m?      

d. Which graph has the smallest value for m?      

#7 Points possible: 5. Total attempts: 5

A cell phone carrier charges a fixed monthly fee plus a constant rate for each minute used.
Part 1. In January, the total cost for 400400 minutes was $82$82 while in February, the total cost for 100100 minutes was $58$58. The constant charge for each minute used is:

· 0.1

· 0.09

· 0.08

Part 2. The fixed montly fee charged by the carrier is;
fee = $ 

–
–

Lesson 3.7

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 12. Simplify algebraic expressions by using the distributive property, combining like terms, and factoring out a greatest common factor.

 

· 13. Evaluate formulas with multiple variables in a variety of contexts, including science, statistics, geometry, and financial math. Solve simple formulas for a specified variable. 

Specific Objectives

· Formulas are relations between variables

· Solving all equations follows the basic rules of undoing and keeping the equation balanced.

Students will be able to

· Evaluate the dependent variable given values for the independent variables.

· Solve for a variable in a linear equation in terms of another variable.

· Factor out a greatest common factor.

In this lesson we will be exploring formulas, which are general relationships between two or more variables.  We have already explored several formulas in this course, and will revisit those as well as explore some new formulas. 

One thing new we will do in this section is solve a formula for a variable.  The process is similar to solving an equation to get a numerical answer, but instead of a number, we get a differently written formula.  To get a feel for how to do this, watch these videos:

·

Solving Formulas – Part 1

 [+]

·

Solving Formulas – Part 2

 [+]

BAC Revisited

Problem Situation 1: The Blood Alcohol Content Formula

Recall that Widmark’s formula for BAC is B=−0.015t+2.84NWgB=-0.015t+2.84NWg   ,  where

B = percentage of BAC

N = number of standard drinks

W = weight in pounds

g = gender constant (0.68 for men, 0.55 for women)

t = number of hours since first drink

For an average male who weighs 190 lbs, the formula can be simplified to

 B=−0.015t+0.022NB=-0.015t+0.022N 

Forensic scientists often use this equation at the time of an accident to determine how many drinks someone had. In these cases, time (t) and BAC (B) are known from the police report. The crime lab uses this equation to estimate the number of drinks (N).

#1 Points possible: 5. Total attempts: 5

For a 190 pound man, find the number of drinks if the BAC is 0.09 and the time is 2 hours.  Give your answer to 1 decimal place.

 drinks

The crime lab needs to use the formula frequently to find the number of drinks N for various 190 pound males who have different values of BAC and hours since drinking.  For example, the next case may involve BAC of 0.11 and time of 3 hours.  They could each time go through the sequence of steps as in question 1.  However, it would be more efficient to solve the formula for the variable N.  That means:  keeping the relationship among the variables the same and the equation balanced, rewrite the formula so that the variable N is alone on one side, while the other side of the equation has variables B and t, and numbers and operations. 

#2 Points possible: 5. Total attempts: 5

Solve the formula  B=−0.015t+0.022NB=-0.015t+0.022N   for NN .  

NN  =    

#3 Points possible: 5. Total attempts: 5

Use the formula for N that you just found to find the value of N when B = 0.09 and t = 2 to verify that the value you get for N is the same as the value found in the first question.  Give your answer to 1 decimal place.

 drinks

#4 Points possible: 5. Total attempts: 5

Use the formula for N that you found in question 2 to find the number of drinks when the 190 pound man has a BAC of 0.11 and the time was 3 hours.  Give your answer to 1 decimal place.

 drinks

Z-Score

Problem Situation 2:  Statistics  z-score Formula

A formula widely used in statistics is  z=x−μσz=x-μσ    ,  where

z = 

z-score

x is a specific data value

μ = mean value      (μ is pronounced “mu”.  It is Greek lower case mu.)

σ = population standard deviation (σ is pronounced “sigma”.  It is Greek lower case sigma.)

It is not necessary that you understand what these terms mean. Rather we will use the formula both to evaluate values and to solve for values.

#5 Points possible: 5. Total attempts: 5

Find the value for z if x = 2.3, μ = 1.9, σ = 0.2

z = 

#6 Points possible: 5. Total attempts: 5

Solve for the value of x if z = 1.5, μ = 1.9, σ = 0.2

x = 

#7 Points possible: 5. Total attempts: 5

Use the z-score formula to solve for the variable x in terms z, μ, and σ.

 xx  =    

To enter μ type mu, and to enter σ type sigma.

Factoring

Factoring

Recall from simplifying expressions the distribution property:

               a(b+c)=ab+aca(b+c)=ab+ac 

Sometimes when solving equations or formulas it can be necessary to use this property in reverse.  When we do this, we call it factoring the greatest common factor (GCF):

               ab+ac=a(b+c)ab+ac=a(b+c) 

Examples:

               x+0.15x=(1+0.15)x=1.15xx+0.15x=(1+0.15)x=1.15x 

               5x+tx=(5+t)x5x+tx=(5+t)x 

               4x+32=4(x+8)4x+32=4(x+8) 

For some more examples, watch this video:

·

Factor GCF

 [+]

This can be used for solving formulas.

Example:  Solve qt+pt=cqt+pt=c  for t.

	(q+p)t=c(q+p)t=c	By factoring out the GCF
	t=cq+pt=cq+p	Dividing both sides by the quantity (q+p)(q+p)

#8 Points possible: 16. Total attempts: 5

Rewrite each of the following expressions by factoring out the Greatest Common Factor

a. 3x+273x+27  =     

b.  12x−812x-8  =     

c. x2−8xx2-8x  =     

d. 4×2+4x4x2+4x  =    

Simple Interest

 

Problem Situation 3:  Simple Interest Formula

The formula A=P+PrtA=P+Prt   allows us to find the amount A in a bank account that has an initial investment P at an annual interest rate of r percent (in decimal form) for t years. When simple interest is used, the interest earned is not paid into the account until the end of the time period.

#9 Points possible: 5. Total attempts: 5

Find the value for A if P = 2000, r = 0.01, and t = 5

A = 

#10 Points possible: 5. Total attempts: 5

Find t if A = 2200, P = 2000, and r = 0.01

t = 

#11 Points possible: 5. Total attempts: 5

Solve for t in terms of A, P, and r.

t =    

#12 Points possible: 5. Total attempts: 5

Solve for P in terms of A, r, and t.   Suggestion:  first factor the right side of the original formula.

P =    

HW 3.7

#1 Points possible: 5. Total attempts: 5

The formula for finding the area of a rectangle is A=LWA=LW .

If the units for length are in feet and the units for width are in feet, what are the units for A? 
    

Solve this formula for L.
L =    

If the length of a rectangle is 13 feet and the width is 6 inches, what is the area of the rectangle in units of square feet? 
    ft2

#2 Points possible: 5. Total attempts: 5

The formula for finding the perimeter of a rectangle is P=2L+2WP=2L+2W .

If the units for length are in feet and the units for width are in feet, what are the units for P? 
    

Solve this formula for L.
L =    
#3 Points possible: 5. Total attempts: 5

The formula for finding the distance traveled, based on speed and time, is D=RTD=RT , where

D is distance
R is rate (speed) 
T is time

Units must be consistent. If the unit for D is miles and the unit for T is minutes, what must the units for R be? 
      

Solve this formula for R.
R =    

If a bicyclist rides for 170 minutes at an average speed of 10 miles per hour, how far was the ride, to 1 decimal place? 
    miles.

At what speed must a bicyclist ride to cover 60 miles in 5 hours, to 1 decimal place? 
    miles/hour.

#4 Points possible: 5. Total attempts: 5

The formula for determining the flow rate of a stream is Q=AVQ=AV , where

Q is the flow rate 
A is the cross sectional area of the stream 
v is the velocity of the water

Units must be consistent. If the units for V are meters per second and the units for A are square meters, what must the units for Q be? 
       

Solve this formula for V.
V =    

A stream has a cross sectional area of 8 square meters. The velocity of the water past this point is 2.6 meters per second What is the flow rate, to 1 decimal place? 
    m^3/s.

What is the speed of the water if the flow rate is 11 m^3/s and the cross sectional area is 5 square feet, to 1 decimal place? 
    meters/second.

#5 Points possible: 5. Total attempts: 5

The formula for determining simple interest is i=prti=prt , where

i is the interest 
p is the principal 
r is the interest rate as a decimal. For example, if the interest rate is 3%, then r = 0.03. 
t is the time

Units must be consistent. If the unit for p is dollars, the unit for i is dollars and the unit for t is years, what must the units for r be? 
      

Solve this formula for r.
r =    

10000 dollars are invested at a simple interest rate of 2 percent for 3 years. How much interest is earned, to the nearest cent? 
    dollars.

What is the interest rate, as a percent, that is necessary to earn 475 dollars in simple interest if 10000 dollars are invested for 1 years, to 1 decimal place? 
    %/year.

#6 Points possible: 5. Total attempts: 5

Acceleration is the rate at which velocity changes. The formula is a=v−v0t−t0a=v-v0t-t0, where

a is acceleration 
v0 is initial velocity (speed), 
v is final velocity 
t0 is initial time, 
t is final time

Units must be consistent.If the units for v and v0 are feet per second and the units for t and t0 are seconds, what must the units for a be?
       

Solve this formula for v.

· v=a−(t−t0)+v0v=a-(t-t0)+v0

· v=a(t−t0)v0v=a(t-t0)v0

· v=at−t0v0v=at-t0v0

· v=a(t−t0)+v0v=a(t-t0)+v0

A car was stopped at a red light. When the light turns green, the car reaches a speed of 55 miles per hour in 4 seconds. What is the acceleration, to 1 decimal place? 
    feet/s2.

What is the final speed of a car that accelerates at 70 ft/s2 for 2 seconds if it’s initial speed was 12 miles per hour, to 1 decimal place? 
    miles/hour.

#7 Points possible: 5. Total attempts: 5

Solve for the specified variable in each equation

a. A group of French students plan to visit the United States for two weeks. They are trying to pack appropriate clothing, but are not familiar with Fahrenheit. One student remembers this formula:
F=95C+32F=95C+32 
where F is the temperature in Fahrenheit and C is the temperature in Celsius. Solve the equation for C.
C =     

b. The simple interest formula is A = P + Prt. In the formula

· A = the full amount paid for the loan

· P = the principle or the amount borrowed

· r = the interest rate as a decimal

· t  = time in years

A car dealership wants to use the formula to find the rate needed for certain values of the other variables.  Solve the formula for r.
r =    

#8 Points possible: 5. Total attempts: 5

Rewrite each of the following expressions by factoring out the Greatest Common Factor

a. 6x+486x+48  =     

b.  15x−1015x-10  =     

c. x2−2xx2-2x  =     

d. 5×2+5x5x2+5x  =    

#9 Points possible: 5. Total attempts: 5

Use Greatest Common Factoring to solve the simple interest formula, A = P + Prt, for P.

P =    
#10 Points possible: 5. Total attempts: 5

Sometimes linear equations are written in forms other than y=mx+by=mx+b .  One common form looks like Ax+By=CAx+By=C .  When you see an equation like this, you can find the slope and y-intercept by solving the equation for y, rewriting it into slope-intercept form.

Rewrite 15x+3y=3015x+3y=30  into slope-intercept form by solving for y.

 y=y=    

From that we can see that:

m =    

b =    

#11 Points possible: 5. Total attempts: 5

Sketch a graph of −6x−4y=−8-6x-4y=-8

Clear All Draw: 

Lesson 4.

1

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 23. Determine the exponential function for a situation when given an initial value and either the growth/decay rate or a second function value. Interpret the initial value and growth rate of an exponential function. Include compound interest as one application.

 

Specific Objectives

Students will understand that

· compounding is repeated multiplication by a compounding factor.

· compounding is best expressed in terms of exponential growth, using exponential notation.

· exponential growth models the compounding of interest on an initial investment.

Students will be able to

· calculate the earnings on a principal investment with annual compound interest.

· write a formula for annual compound interest.

· compare and contrast linear and exponential models.

Certificates of Deposit

A Certificate of Deposit (CD) is a type of investment used by many people because it is very safe and predictable. CDs can be purchased through banks and other financial institutions. The U.S. Security and Exchange Commission gives the following information about CDs.

The ABCs of CDs

A CD is a special type of deposit account with a bank or thrift institution that typically offers a higher rate of interest than a regular savings account. Unlike other investments, CDs feature federal deposit insurance up to

$

250,000.

When you purchase a CD, you invest a fixed sum of money for fixed period of time—six months, one year, five years, or more—and, in exchange, the issuing bank pays you interest, typically at regular intervals. When you cash in or redeem your CD, you receive the money you originally invested, plus any accrued interest. If you redeem your CD before it matures, you may have to pay an “early withdrawal” penalty or forfeit a portion of the interest you earned …

At one time, most CDs paid a fixed interest rate until they reached maturity. But, like many other products in today’s markets, CDs have become more complicated. Investors may now choose among variable-rate CDs, long-term CDs, and CDs with other special features.

In this course, you will only be looking at situations in which the CD has a fixed interest rate. The period of time that you agree to leave your money in the CD is called the term. Terms can range from 3 months to multiple years. Some CDs require a minimum deposit, but others do not. The deposit, or amount you put into the CD, is called the principal.

CDs pay compound interest. Recall in previous assignments you have worked with situations that use simple interest. In simple interest, the total interest is calculated from the original principal. In compound interest, the interest is added to the principal after a set period of time. Then interest for the next period of time is calculated based on the new, higher balance in the account.

Here is an example:

You invest $100 at 5% interest compounded annually or each year. At the end of the year, you earn 5% of $100 or $5. This is added into the account so now the balance is $105. In the second year, you earn 5% of $105 or $5.25.

Following are some common compounding periods:

·

Quarterly

(4 times a year)

·

Monthly

(12 times a year)

·

Daily

(365 times a year)

There is also a type of compounding called continuous, but that will not be discussed in this course.

Five-year CD

Problem Situation 1: The Five-year CD

Suppose you invest $1,000 principal into a certificate of deposit (CD) with a five-year term that pays a 2% annual percentage rate (APR) interest. The compounding period is one year.

To calculate how much you have after a year:

Interest = Principal * interest rate.         I=PrI=Pr  

End amount = Principal + Interest.        A=P+IA=P+I 

Putting these together:                           A=P+PrA=P+Pr 

#1 Points possible: 5. Total attempts: 5

How much money will you have in your account after the first year?  Round to the nearest cent if necessary.

$

#2 Points possible: 12. Total attempts: 5

How much will you have at the end of the five-year term?  Calculate the balance after each year (you already found the balance after 1 year above)  Round to the nearest cent if necessary.

Year

Account Balance

 

2 years

$

3 years

$

4 years

$

5 years

$

For the first year, you found the balance using the calculation 1000+1000(0.02)1000+1000(0.02) 

Notice that 1000+1000(0.02)=1000⋅1+1000(0.02)1000+1000(0.02)=1000⋅1+1000(0.02)  and we can factor out the Greatest Common Factor of 1000, so

1000⋅1+1000(0.02)=1000(1+0.02)=1000(1.02)1000⋅1+1000(0.02)=1000(1+0.02)=1000(1.02) 

In year 2, you calculated:  1020+1020(0.02)1020+1020(0.02) .  We can factor out the GCF of 1020, so

1020+1020(0.02)=1020⋅1+1020(0.02)=1020(1+0.02)=1020(1.02)1020+1020(0.02)=1020⋅1+1020(0.02)=1020(1+0.02)=1020(1.02) 

Now, remember where that 1020 came from:  1000+1000(0.02)=1000(1.02)=10201000+1000(0.02)=1000(1.02)=1020 .  

If we take the expression for the balance after 2years,  1020(1.02)1020(1.02) , and replace the 1020 with 1000(1.02)1000(1.02)  then we find another way to calculate the balance after 2 years:  1020+1020(0.02)=1020(1.02)=1000(1.02)(1.02)1020+1020(0.02)=1020(1.02)=1000(1.02)(1.02) 

#3 Points possible: 5. Total attempts: 5

The expression 1000(1.02)(1.02)1000(1.02)(1.02)  could be written more compactly by using exponents.  Rewrite the expression using exponents.

   

#4 Points possible: 5. Total attempts: 5

The amount in the account after 3 years could be calculated as  1040.40+1040(0.02)1040.40+1040(0.02)  or  1040.40(1.02)1040.40(1.02) .  

Or, we could use the same approach shown above and replace the 1040.40 with the expression for the value after 2 years you found in the previous problems.  Make that replacement, then try to write it more compactly using exponents.

   

Hint: If you get stuck, we’ll offer some hints

#5 Points possible: 5. Total attempts: 5

Look at the simplified expressions you have found for the amount after two and three years.  Do you notice a pattern in how those expressions look?

Use that pattern to develop a formula for the account balance after t years.

A =    

Hint: If you get stuck, we’ll offer some hints

#6 Points possible: 20. Total attempts: 5

Using the formula you developed in the previous problem, fill in the table below for the account balance after the given number of years, if the principal = $1,000 and the interest rate APR = 2%.  Round to the nearest cent if needed.

Year

Account Balance

 

10 years

$

20 years

$

40 years

$

70 years

$

100 years

$

#7 Points possible: 5. Total attempts: 5

Plot your results from the previous question. You won’t be able to plot the exact values, so you will need to estimate where the values will lie.

Clear All Draw: 

#8 Points possible: 5. Total attempts: 5

Is your formula you developed earlier for the account balance linear?  If not, what family does it belong to?

· Linear

· Exponential

· Quadratic

#9 Points possible: 5. Total attempts: 5

In an earlier problem, you came up with a formula that gave the account balance after t years.  Look carefully at that formula, and consider where the numerical values in the formula came from.

Write a general formula that could be used to find the accrued amount(A) for a CD with annual compounding.

Let

 P = the principal, r = the APR as a decimal, and t = number of years.

A =    

Value of a CD

Problem Situation 2: The Value of a CD

In the last problem, the CD was compounded annually. In this problem, we will extend that for compounding periods of various durations.

Suppose you invest $1,000 principal in a two-year CD, advertised with an annual percentage rate (APR) of 2.4%, where compounding occurs monthly.

#10 Points possible: 8. Total attempts: 5

If the APR is 2.4% per year, how much is the interest per month (Remember Periodic Rate from 1.9)?

 %

Now write this as a decimal.

#11 Points possible: 24. Total attempts: 5

Using your answer from the previous problem, and the concepts developed earlier in the lesson, complete the following table.  Round values to the nearest cent if needed.

Period

Account Balance

1 month

$

2 months

$

6 months

$

12 months

$

24 months

$

3 years

$

Hint: If you get stuck, we’ll provide some hints

In Question 10, you figured out the periodic rate (interest per month) by dividing the APR by 12, the number of times the interest is being compounded in one month.  In Question 11, to find the account balance after 24 months, you used 24 months in the exponent.  

In the next problem, we want to develop a general formula to calculate the value of any CD.  

Let

P = the principal, 
r = the annual interest rate (APR) as a decimal,
n = the number of compounding periods in a year (1 for annual, 12 for monthly, 52 for weekly, etc.)
t = number of years.

So in the previous two questions, n = 12, since we were compounding monthly (12 times per year).  Think about how you answered the last question:  how did you find the interest rate per period?  To find the value after 3 years, how did you figure out the number of months?

#12 Points possible: 5. Total attempts: 5

Write a general formula that can be used to calculate the value of any CD, using the variables defined above (P, t, r, and n).

A =    
Hint: If you get stuck, we’ll provide some hints

HW 4.1

#1 Points possible: 5. Total attempts: 5

Which of the following was one of the main mathematical ideas of the lesson?

· Compound interest is an important concept in savings and investment.

· Exponential equations can be used to model percentage growth.

· Compound interest can be modeled with linear equations.

·

Annual

compounding is done once a year.

#2 Points possible: 1. Total attempts: 5

Explain what the following statements mean. Give examples to support your explanation.

· Multiplication is repeated addition.

· A power is repeated multiplication.

#3 Points possible: 1. Total attempts: 5

How is a linear model different from an exponential model?

#4 Points possible: 24. Total attempts: 5

In Lesson 3.7, you explored the Simple Interest formula, A=P+PrtA=P+Prt , which is used when interest does not compound.  This is fairly rare, but is how Treasury Notes and some other specialized investments work.

In this problem we will compare Simple Interest to Compound Interest, which you learned in this Lesson.  Recall from the lesson that when interest is compounded annually the compound interest formula is A=P(1+r)tA=P(1+r)t.

In both formulas, A is the 
Amount
 (total principal plus interest) you end up with, P is the 
Principal (starting amount), r is the 
annual interest rate
 (quoted as a percent, but used as a decimal), and t is the 
time
 in years.

a. Complete the table below to compare the amounts you would have if you invested $1000 at 5% with simple interest compared to $1000 at 5% compounded annually.  Round to the nearest cent.

 

Simple Interest

Compound Interest

1 year

$

$

2 years

$

$

3 years

$

$

4 years

$

$

5 years

$

$

b.
 

c. In the lesson you noticed that compound interest is an exponential model.  What kind of model is simple interest? 
     

d. In the long run, which method of receiving interest will result in more money being in the account, assuming the rate of interest is the same?  
   

#5 Points possible: 12. Total attempts: 5

Which is the better investment for 10 years: investing $500 at 10% APR or $1,000 at 2% APR? Both investments have annual compounding.

After 10 years, $500 at 10% APR would grow to $

After 10 years, $1,000 at 2% APR would grow to $

The better investment is:    

 

CDs are a very safe investment because they are usually insured by the U.S. government. (There are some CDs that are not insured so it is important to always check!) Because they are so safe, CDs earn low rates of interest. The amount earned on an investment is often called the return. In general, investments with higher risk also have the potential for higher rates of return. For example, if you invest in a stock, which is like buying a piece of a company, you might earn far more than you would with a CD. However, you also run the risk of losing all of your money.

Mutual funds are another type of investment used by many people. The U.S. Security and Exchange Commission defines a mutual fund as follows:

1

A mutual fund is a company that brings together money from many people and invests it in stocks, bonds, or other assets. The combined holdings of stocks, bonds, or other assets the fund owns are known as its portfolio. Each investor in the fund owns shares, which represent a part of these holdings.

Mutual funds are an attractive investment to many people for several reasons.

· You can invest small amounts of money at a time.

· The fund is managed by a company that does all the work of researching and choosing specific investments.

· In general, there is less risk than owning a single stock because the money is spread across many investments.

· In general, there are higher rates of returns than those available in insured investments like CDs and savings accounts.

However, it is important to understand that mutual funds do still have risks. It is possible to lose some or all of your investment. Also, funds charge fees to pay for the management. Sometimes these fees can be very high.

Selecting investments is an important decision and should be researched carefully. Through this course, you will be introduced to a few basics concepts related to investment, but you should not make any decisions based only on the information presented here.

One of the most important concepts in investing is to take advantage of the power of compounding over time. The following example will help you explore this idea.

 

#6 Points possible: 16. Total attempts: 5

Lorenzo

and

Michael

each decide to invest in a mutual fund to save for retirement. They each choose a mutual fund that has had an average annual return of 5.6% over the last decade. There is no guarantee that the mutual fund will continue to earn this same rate, but it can be used as an estimate of future returns. Use the information given below to estimate how much each man will have when he retires. Round to the nearest dollar.

· Lorenzo invests $2,000 when he is 25 years old.

· Michael invests $5,000 when he is 45 years old.

· Both men plan to retire at age 65.

Hint 1: The amount earned is how much they have beyond the initial investment.
Hint 2: You can treat this problem as if the account compounds annually.

 

$

$

$

$

Amount at retirement	Amount earned
Lorenzo
Michael

#7 Points possible: 5. Total attempts: 5

In this problem you will compare the effects of different compounding periods on the interest an investment earns. Complete the table below using the values indicated. Show the formula you used, with the correct values, in the second column.  In the third column, give the result, rounded to the nearest cent.

· Principal: $1,000

· APR: 4.5%

· Time: 10 years

A =    

$

A =    

$

A =    

$

A =    

$

Compounding Period	Equation Used for Calculation (with values inserted)	Amount Accrued  After 10 Years
Annual
Quarterly
Monthly
Daily

Lesson

3

.5

Introduction

Course Objectives

This lesson will address the following course outcomes:

· 20. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).

 

 

· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.  

· 25. Use functional models to make predictions and solve problems. 

Specific Objectives

Students will understand that

· linear equations can approximate nearly linear data.

Students will be able to

· find the equation of a line that estimates nearly linear data by calculating the rate of change and the vertical intercept of the line.

· use approximate linear models to interpolate and extrapolate.

Equation of Line Between Two Points

In an earlier lesson, you learned that the equation of line can be written in the form y=mx+by=mx+b .  This makes it easy to write the equation when we know the slope and initial value.

In the last lesson, you learned how to compute the slope when you had two points that lie on the line, using the formula slope=change of outputchange of input=y2−y1x2−x1slope=change of outputchange of input=y2-y1x2-x1 .

Now suppose had two points, neither of which was the y intercept, and we want to write an equation for the line.  To do that, we need to use the slope and a point to find the intercept.

Example: Find the equation of a line passing through (2,5) and (4, 6).

The slope is change of outputchange of input=y2−y1x2−x1=6−54−2=12change of outputchange of input=y2-y1x2-x1=6-54-2=12 

We know the general equation of a line is , so putting in the known slope gives

 y=12x+by=12x+b 

Now, to find b we can replace x and y with one of our points, and solve for the value of b that will make the equation true.  Using (2,5):

5=12(2)+b5=12(2)+b 

5=1+b5=1+b 

4=b4=b 

We have found the y-intercept and can now write the equation for the line:  y=12x+4y=12x+4 

The process for finding an equation of a line is:

1. Identify two points that lie on the line

2. Calculate the slope between those points.  m=y2−y1x2−x1m=y2-y1x2-x1 

3. Start with y=mx+by=mx+b .  Put in the slope, substitute one of the points for x and y, and solve for b.

4. Write the final equation of the line.

For more about finding the equation of line, here are some videos:

·

Finding the equation of a linear in slope-intercept form.
 [+]

·

Another example of finding the equation of a line given two points.

 [+]

#1 Points possible: 5. Total attempts: 5

Find an equation of line through the points (15,8) and (25,6). Use y for the output, and x for the input.

   

#2 Points possible: 5. Total attempts: 5

Recall in the last lesson that Raj weighed 193.4 pounds after 0.5 weeks, and 187.6 pounds after 4 weeks.  Find a linear model for Raj’s weight, w, after t weeks.  Round the slope and intercept to two decimal places if needed.

  

 

Minimum Wage

Problem Situation 1: Minimum Wage

The minimum wage is the lowest amount a company is allowed to pay its workers.  In the United States, the minimum wage was introduced in 1938.  The federal government sets a minimum wage, currently

$

7.25 per hour, and states and cities can set a higher minimum wage if desired.  In Washington State in

2015

, the minimum wage was

$9.47

per hour, and is adjusted each year based on cost of living.

As you may have read in the newspapers, there are groups working to raise the minimum wage to $12 or $15.  Seattle has adopted a plan to increase the minimum wage to $15 over the next several years (by 2017 or 2018 for large employers, and by 2019 or 2021 for smaller companies).  How much of a jump is this over the normally expected increases?

To explore this question, we will look at the historical increases of minimum wage in Washington. Real data rarely fall on a straight line, but sometimes data show a definite trend. If the trend is close to linear, the data can be approximated by a linear model. This means that a linear model gives good estimates of what the data will be if the trend continues. A model can also be used to estimate values between data points.In this lesson, you will learn to create linear models from data.

The following data shows the minimum wage in Washington from

2005

to 2015. 

$8.55

	Year	2005	2006	2007		2008		2009	20 10	2011	2012	2013	2014	2015
Min Wage	$7.35	$7.63	$7.93	$8.07		$8.55	$8.67	$9.04	$9.19	$9.32	$9.47

 

When data appears approximately linear, we might want to find a linear model to approximate the data.

#3 Points possible: 4. Total attempts: 5

Each of the lines below was found by using a pair of values from the data.  Which of the lines is a reasonable approximation of the data (there may be more than one answer)?

 
·
·
·
·

#4 Points possible: 5. Total attempts: 5

Find an equation for the linear model approximating the minimum wage data, using the data values from 2005 and 2015, letting W be the minimum wage and t the year, measured in years after 2000.  Round values to three decimal places if needed.

   

#5 Points possible: 15. Total attempts: 5

Fill in the blanks below to explain the meaning of the slope and intercept of your equation.
The equation tells us that       approximately  and it has been     by       

#6 Points possible: 5. Total attempts: 5

Using your model, predict the minimum wage in 2020. 

$

#7 Points possible: 8. Total attempts: 5

When does your model predict the minimum wage would reach $15 if it continued increasing at the same rate?

Algebraically, it would happen when t =   (give this answer to one decimal place)

Practically, this means in the year   (give this to a whole year)

Height Chart

Problem Situation 2: Height Chart

The chart to the below is a growth chart for boys, aged 2 to 15 years.  The different curves show different percentiles for growth; the top curve shows the 95th percentile, where 95% of boys are that height or shorter.  The middle curve shows the 50th percentile (the median height), and the bottom curve shows the 5th percentile.

#8 Points possible: 5. Total attempts: 5

Find an equation for an approximate linear model for the median height curve, using the data values from ages 3 and 15.  Write the height, H, in terms of the age, t. 

   

#9 Points possible: 10. Total attempts: 5

Use your model to predict the height of a 10 year old boy. 

 cm

How well does your model’s prediction agree with the data in the chart?

· The value is not very close

· The value is very close

#10 Points possible: 10. Total attempts: 5

Use your model to predict the height of a 16 year old boy.

 cm

Are you more or less confident in this prediction than the prior one?

· More confident

· Same confidence

· Less confident

#11 Points possible: 10. Total attempts: 5

Use your model to predict the height of a 30 year old man.

 cm

How accurate is this prediction likely to be?

· Very accurate

· Somewhat accurate

· Not accurate

#12 Points possible: 8. Total attempts: 5

Determine a reasonable domain for this model.  That is, determine an interval of ages for which this model is likely to be reasonably accurate.

 ≤ age ≤ 

#13 Points possible: 8. Total attempts: 5

Ninety percent of children fall between the 5th percentile and the 95th percentile. We’ll call this the “typical” interval.  Using the chart, write an inequality for the “typical” interval of heights for a 9 year old boy.

 cm ≤ height ≤  cm

#14 Points possible: 8. Total attempts: 5

If a lost male child is found who is 145 cm tall, determine an interval of likely ages for the child.  Give whole numbers.

 ≤ age ≤ 

Summary – Finding Linear Formulas

In the last several lessons you have explored linear equations in a variety of ways.  It can feel like there is a lot of different things to remember, but they’re really all related – each lesson we’ve been adding one new layer to what was done in the previous lesson.  Here is a summary to tie together everything you’ve learned in the last several lessons.

Linear Equations

Linear equations have the form y=mx+by=mx+b , where

· m is the rate of change, or slope.

· The units will be “outputs per input”.  For example if the input is years and outputs is people, the slope will have units “people per year”

· b is the initial value, or vertical-intercept

· The units will be the same as the output quantity.  For example if the input is years and outputs is people, the vertical intercept will have units “people”.

To find a linear equation from given information:

· Is the slope (rate of change) given to you?  

· If yes, identify it

· If no, calculate it from the given data, using m=change of outputchange of input=y2−y1x2−x1m=change of outputchange of input=y2-y1x2-x1 

· Is the vertical intercept given to you?

· If yes, identify it

· If no, plug any given (x,y) point and the slope into the equation y=mx+by=mx+b  and solve for b.

· Now that you have the slope and intercept, write the equation.

For examples of all the cases, 

see this video

 [+]

HW 3.5

#1 Points possible: 5. Total attempts: 5

Which of the following was one of the main mathematical ideas of the lesson?

· It is necessary to be given the starting value of a linear model or it is impossible to find the equation

· It is necessary to have three or more points to find a linear model

· Printing costs can be modeled with a linear equation

· Given the slope and a point, you can solve for the vertical intercept of a linear model

The flight times and distances from Memphis, Tennessee, to various cities are given in the table and graph below. Use this information to answer Questions 2–4.

Time to Fly from Memphis to Various Cities
City	Distance (Mi)	Time (minutes)
Atlanta	332	82
Boston	1139	191
Detroit	611	121
Baltimore	787	162
Cincinnati	403	100

 Flight times retrieved for flights leaving July 28, 2011 from Delta.com on July 22, 2011.

#2 Points possible: 5. Total attempts: 5

Use what you know about the relationship of the equation to the graph to select the model from the list below that is most representative of the data. Do not actually find a linear model to make your choice.

· t = -0.13d + 110 with d representing the distance in miles and t representing the time in minutes.

· t = 0.13d + 40 with d representing the distance in miles and t representing the time in minutes.

· t = -0.13d + 40 with d representing the distance in miles and t representing the time in minutes.

· t = 0.13d + 110 with d representing the distance in miles and t representing the time in minutes.

#3 Points possible: 6. Total attempts: 5

Using the model you chose in the previous problem, estimate the flight time from Memphis to Philadelphia if the distance between the two cities is 874 miles.  Round to the nearest minute.

 hours and  minutes

#4 Points possible: 5. Total attempts: 5

Based on the model you chose, select the response that best tells how long it will take a Delta plane once in the air flying at full speed to travel 1 mile. 
Hint: The model includes both flying time and time on the ground.

· It takes 0.13 minutes or around 8 seconds to go one mile.

· It takes 40.13 minutes to go one mile.

· The plane goes 0.13 miles in one minute so it takes between 7 and 8 minutes to go one mile.

· It takes 40 minutes to go one mile.

The table below shows the world record in the men’s 100-meter dash from

1912

to 2009. Use this to answer Questions 5–8. (Note: A change to electronic timing in the 1970s might explain the 30-year gap in records from

1968

to

1999

.)

Year

2008

2009

Men’s 100=meter dash 2
World Record Time  (nearest tenth)
1912	10.6
1921	10.4
1930	10.3
1936	10.2
1956	10.1
1960	10
1968	9.9
1999	9.8
9.7
9.6

 

#5 Points possible: 5. Total attempts: 5

Create a scatter plot on paper – it’s best to do this on graph paper – using the data relating year and world record time on the 100 meter dash. Let the horizontal axis represent years, t, where t is the number of years after 1900.  The vertical axis represents R, the time in seconds of the record 100 meter dash.  Use a straight edge to draw the line you think is the best linear model for this data.  Write the equation of that line here.

R =    

#6 Points possible: 5. Total attempts: 5

Use the model you created to predict when the world record will fall below 9.0 seconds.

The model predicts the world record will fall below 9.0 seconds in year 

#7 Points possible: 12. Total attempts: 5

Identify whether the following statements are true or false based on your model. Note that the values given are approximations.

a. On average, the men’s world record in the 100-meter dash has decreased about one-hundredth of a minute each year. 
    

b. On average, the men’s world record in the 100-meter dash has decreased about one-hundredth of a second each year. 
    

c. On average, the men’s world record in the 100-meter dash has decreased about one-tenth of a second every 10 years. 
   

#8 Points possible: 8. Total attempts: 5

Models often only work well for a limited range of input values. Outside that range, the model is said to break down. Which of these explains why your world record model might break down? There may be more than one correct answer.

· Eventually the model will predict negative times for the race.

· The times are not really accurate.

· Changes in equipment or training might make for sudden improvements in times and change the trend in the current data.

· The data do not form a perfect line.

#9 Points possible: 5. Total attempts: 5

The following chart gives sunrise times for New York City in hours after midnight as measured in Eastern Standard Time (EST) on the 15th of each month. The first month is August 2010, the second month is September 2010, and so on. (Note: Since you are looking at general trends, scales are not included on this graph.)

Would it be appropriate to use a linear model to represent these data?     
Explain.

In baseball, a team must score more runs than its opponent to win the game. To score runs players must reach base. This is measured by the on-base percentage. A player who reaches a base may or may not score. The following graph compares a team’s on-base percentage with the runs scored. The scatterplot gives data relating the number of runs and the on-base percentage. Use the graph to answer Questions 10 and 11.

#10 Points possible: 5. Total attempts: 5

Katy used the graph to create a linear model that she could use to predict the number of runs scored using the on base percentage. Her model was r = 30b + 450 where, r = number of runs and b = on-base percentage  (e.g., for 32%, b=32). What is wrong with Katy’s model?

· 450 runs

is not the vertical intercept because the scale on the horizontal axis does not start at 0.

· Katy put the rate and the vertical intercepts in the wrong places in the equation.

· The data are not nearly linear so a linear model should not be used.

#11 Points possible: 5. Total attempts: 5

Based on the slope in Katy’s model, r = 30b + 450, if on-base percentage for a team goes up by 1% then by about how much does the number of runs scored go up?

· 450 runs

· 0.03 runs

· 30 runs

#12 Points possible: 20. Total attempts: 5

The graph below shows the cost of basic cable service from a cable provider from 2002 to 2010.3

a. A representative of the company made a linear equation of the data using the 2002 price as a starting value and the 2005 price as a second data point. What is the company’s equation? Use Pc for the price per month and t for time in years since 2002. Round the slope to the nearest hundredth.
    

b. A consumer advocate made a linear equation of the data using the 2002 price as a starting value and the 2009 price as a second data point. What is the advocate’s equation? Use Pa for the price per month and t for time in years since 2002. Round the slope to the nearest hundredth.
    

c. What will the price per month be in 2015 based on the company’s model? The consumer advocate’s model? Hint: Remember that t is the number of years since 2002. 
Company’s projection: $
Consumer advocate’s projection: $

d. Why might the company and the consumer advocate choose those particular points to make their models?

e. Which of the following is the most accurate estimate of the relative increase from 2002 to 2010?

· 90-100%

· 75-85%

· 50-60%

· 35-45%

#13 Points possible: 5. Total attempts: 5

Find the equation (in terms of xx) of the line through the points (-2,-9) and (5,5)
yy =    

#14 Points possible: 5. Total attempts: 5

Find the equation (in terms of xx) of the line through the points (-4,3) and (1,5)

y=y=   

#15 Points possible: 5. Total attempts: 5

A company has a manufacturing plant that is producing quality canisters. They find that in order to produce 130 canisters in a month, it will cost $3840. Also, to produce 370 canisters in a month, it will cost $8160. Find an equation in the form y=mx+b,y=mx+b, where xx is the number of canisters produced in a month and yy is the monthly cost to do so.
Answer: y=y=