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Linear Models, Break Even, Rate of Change

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1. A tree increases in diameter each year by an average of 0.51cm. If the tree currently has a diameter of 4 inches, how old do you estimate the tree is?

2. If rain is falling at a rate of ¼ inch per hour, how much rain would you expect after 6 hours?

3. Consider the variation of temperature with altitude depicted in the graph below.

a. What is the dependent variable?
b. What is the independent variable?
c. What is the "y-intercept"?
d. What is the slope?
e. Write a sentence to explain what the slope means.
f. Use your answers to construct an equation for the linear model, y = mx + b, that describes the graph.
g. Use your model to predict what the temperature would be at 7000 feet above sea level.

4. The fair market rental rate for Columbus was $730 in 2013 and $551 in 2003. Find and interpret the rate of change for rent in Columbus during this period.

5. The environment club has saved 200 pounds of cans and promises to collect 20 pounds of cans each week and will use the profits to buy additional recycling bins. Construct a linear model for their collection program.
a. What will the independent variable, x, represent?
b. What will the dependent variable, y, represent?
c. What is the starting value, b, for this model?
d. What is the slope, m, for this model?
e. Write the model in y = mx + b form.
f. How many pounds of cans will the club have by week 12?
g. When will the club have 320 pounds of cans?

6. The psychology club is having a self-proclaimed psychic come to their campus fund-raising event to demonstrate his abilities. He charges $200 for these events, and the club is charging $1.50 for tickets with a chance to have the psychic read the ticket holder's mind.
a. Let x represent the number of tickets sold and y represent money in a linear model which can be used to estimate the profit or loss based upon the number of tickets sold.
b. How much would the club gain or lose by selling 125 tickets?
c. How many tickets are needed to break even?

7. The cost of a LCD TV dropped from $265 in 2012 to $199 in 2014.
a. Find the rate at which the cost has been decreasing.
b. Construct a linear model to estimate future costs of TV's.
c. Use your model to estimate what the cost will be in 2016.

8. Online jewelry sales have increased steadily over the last 10 years. In 2003, sales were approximately 2 billion dollars, and in 2010 they were approximately 10 billion.
a. Find the rate at which online jewelry sales have been increasing.
b. Construct a linear model which can be used to estimate jewelry sales.
c. Use your model to estimate online jewelry sales for 2014.

9. In 1963, postage was 5 cents per ounce. In 1981, postage was 18 cents per ounce.
a. Establish a linear model which can be used to estimate the postage per ounce based upon the year (or you can use the number of years since 1963.)
b. If the trend had continued through to today, what would the postage per ounce be?

10. Actual price of postage today is 46 cents for the first ounce and 20 cents for each additional ounce.
a. Construct a linear model which can be used to estimate cost of postage.
b. Use your model to estimate the postage of a package which weighs 250 grams.
c. If you are willing to spend no more than three dollars on postage for a package, what is the most a package can weigh that you could send on budget?

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Solution Summary

Step-by-step solutions are provided to the various questions about linear models.

Solution Preview

1
1 inch = 2.54 cm
4 inches = 10.16 cm
10.16 / 0.51 = 19.9
20 years old

2
1/4 X 6 = 6/4 = 3/2
3/2 inches

3a
temperature
3b
altitude
3c
60.0
3d
(60 - 42) / (0 - 5000) = 18/(-5000) = -0.0036
3e
The temperature decreases by 0.0036F for every 1 feet above sea level.
3f
y = -0.0036x + 60.0
3g
y = -0.0036 * 7000 + 60.0 = 34.8

4
(730 - 551) / (2013 - 2003) = 17.9
The fair market ...

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  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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