Applications of Systems of Linear Equations

Renting a car

Question 1. You are going on a road trip and want to rent a car. You need to decide which agency to rent a car from. Agency A charges \$50 plus 10 cents (0.10 dollars) for each mile you drive. Agency B charges \$30 plus \$0.20 per mile. Agency C charges \$0.50 for each mile you drive. The cost in dollars for renting a car for one day from these three different rental agencies and driving it $d$ miles is given by the following equations:

$A=50+0.10d$
$B=30+0.20d$
$C=0.50d$

Graphs of the cost equation for each of the three agencies are shown on the grid to the left.

If you drive 150 miles, how much will each agency charge you? (Click somewhere on the $d=150$ gridline, and the computer will calculate the answer for you on the right hand side of the grid.)	Agency A \$
	Agency B \$
	Agency C \$

If you are only planning on traveling 80 miles, which agency should you rent a car from?

How many miles do you have to drive for the cost of renting from agency A and agency C to be the same?

When is it cheapest to rent from agency B?

Between and miles

If you are planning on driving 400 miles, which agency should you rent a car from?

Cost, revenue, and profit

Question 2. A company that manufactures and sells running shoes has a fixed overhead cost of \$650,000. It costs the company an additional \$20 to produce each pair of running shoes and it sells each pair of shoes for \$70.

Write an equation to describe $C$, the total cost to the company in terms of $n$, the number of pairs of shoes it produces. (It costs the company \$650,020 to produce one pair of shoes, \$650,040 to produce 2 pairs of shoes, \$650,060 to produce 3 pairs of shoes, etc.)

$C=$

Write an equation that describes the total revenue, $R$, in terms of the number of pairs of shoes the company sells. (The revenue of the company is its income from selling shoes.)

$R=$

Question 3. Click on . Then slide the values of $a$ and $b$ so that the equations for $C=650000+an$ and $R=bn$ match the equations you found. Click inside the grid and use the mouse or the left and right arrow keys to answer the following questions.

Does the company make or lose money if it sells 5000 pairs of shoes?

What is the cost of making 5000 pairs of shoes?

What is the revenue if the company sells 5000 pairs of shoes?

How many pairs of shoes should the company sell in order for the cost and revenue to be the same?

What is the company's profit if it sells 20,000 pairs of shoes?

What would the equation for $R$ be if the company sold each pair of shoes for \$150? Use the slider for $b$ to change the equation for $R$ to the equation you found.

How many pairs of shoes would the company have to sell before it could make a profit if it sold each pair for \$150?

What would the equation for revenue be if the company sold each pair of shoes for \$20?

Use the slider for $b$ to change the equation for $R$ to the equation you found. What do you notice about the two lines?

Would the company be able to make a profit at this price? Why or why not?

Which calling card should you use?

Question 4. Supertel Communications offers two different calling cards: the Platinum Card and the Gold Card. Every time you use the Platinum Card you pay a 50¢ connection fee and 3¢ for every minute that you talk on the phone. The Gold Card has no connection fee, but you pay 7¢ for every minute of use. If $t$ is the number of minutes you speak on the phone, $P$ the cost in cents of using the Platinum Card and $G$ the cost of using the Gold Card, write equations relating $P$ and $G$ to $t$.	Platinum Card
	Gold Card

Question 5. Click on to see the example for question 5. Slide the values of $a$, $b$, and $c$ so that the equations $P=a+bt$ and $G=ct$ match the equations you found in Question 4.

What does the point of intersection of the graphs of $P$ and $G$ represent?

If you plan to talk on the phone for 15 minutes, should you use the Platinum or the Gold card?

What if you want to talk on the phone for 5 minutes?

Birth and death rates

Question 6. The tables below show the crude birth and death rates per 1000 people in France for the years 1975-1998. Click on to see the data from these tables plotted on a grid.

Birth Rate

1975	1980	1985	1990	1994	1998
14.1	14.8	13.9	13.5	12.3	11.68

Death Rate

1975	1980	1985	1990	1994	1998
10.6	10.2	10.1	9.3	9.0	9.12

There are two lines which you will move to try to find a good “fit” to the data from the tables. Look to the right of the grid. The numbers 1.946 and 0.9597 are called the root-mean-square errors of the lines. The root-mean-square error tells you how far away a line is from the data (a smaller number means the line is a better “fit” to the data).

Alternate sliding the values of $m$ and $n$, as well as $p$ and $q$, until you find approximately the best equation for each set of data. Your root-mean-square error should be less than 0.49 for $B=mt+n$ (the birth rate) and less than 0.19 for $D=pt+q$ (the death rate). (On the vertical axis you see the rate of birth and death per 1000 people. On the horizontal axis you see the number of years since 1975.)

What equation did you find for $B$, the birth rate?

$B=$

What equation did you find for $D$, the death rate?

$D=$

Approximately when will the birth and death rates be the same in France? (Remember that $t$ is the number of years since 1975.)

If no people move in or out of France, will the population increase or decrease after this year? Why?

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