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# 2.3.3E: Modeling with Linear Functions (Exercises)


section 2.3 exercise

1. In 2004, a school population was 1001. By 2008 the population had grown to 1697. Assume the population is changing linearly.

a. How much did the population grow between the year 2004 and 2008?
b. How long did it take the population to grow from 1001 students to 1697 students?
c. What is the average population growth per year?
d. What was the population in the year 2000?
e. Find an equation for the population, P, of the school t years after 2000.
f. Using your equation, predict the population of the school in 2011.

2. In 2003, a town’s population was 1431. By 2007 the population had grown to 2134. Assume the population is changing linearly.

a. How much did the population grow between the year 2003 and 2007?
b. How long did it take the population to grow from 1431 people to 2134?
c. What is the average population growth per year?
d. What was the population in the year 2000?
e. Find an equation for the population, $$P$$, of the town $$t$$ years after 2000.
f. Using your equation, predict the population of the town in 2014.

3. A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be$118.

a. Find a linear equation for the monthly cost of the cell plan as a function of $$x$$, the number of monthly minutes used.
b. Interpret the slope and vertical intercept of the equation.
c. Use your equation to find the total monthly cost if 687 minutes are used.

4. A phone company has a monthly cellular data plan where a customer pays a flat monthly fee and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be$17.80.

a. Find a linear equation for the monthly cost of the data plan as a function of $$x$$, the number of MB used.
b. Interpret the slope and vertical intercept of the equation.
c. Use your equation to find the total monthly cost if 250 MB are used.

5. In 1991, the moose population in a park was measured to be 4360. By 1999, the population was measured again to be 5880. If the population continues to change linearly,

a. Find a formula for the moose population, $$P$$.
b. What does your model predict the moose population to be in 2003?

6. In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. If the population continues to change linearly,

a. Find a formula for the owl population, $$P$$.
b. What does your model predict the owl population to be in 2012?

7. The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010, and is being depleted by about 2.1 billion cubic feet each year.

a. Give a linear equation for the remaining federal helium reserves, $$R$$, in terms of $$t$$, the number of years since 2010.
b. In 2015, what will the helium reserves be?
c. If the rate of depletion doesn’t change, when will the Federal Helium Reserve be depleted?

8. Suppose the world’s current oil reserves are 1820 billion barrels. If, on average, the total reserves is decreasing by 25 billion barrels of oil each year:

a. Give a linear equation for the remaining oil reserves, $$R$$, in terms of $$t$$, the number of years since now.
b. Seven years from now, what will the oil reserves be?
c. If the rate of depletion isn’t change, when will the world’s oil reserves be depleted?

9. You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 $$plus$$ 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable? 10. You are choosing between two different window washing companies. The first charges$5 per window. The second charges a base fee of $40 plus$3 per window. How many windows would you need to have for the second company to be preferable?

11. When hired at a new job selling jewelry, you are given two pay options:

Option A: Base salary of $17,000 a year, with a commission of 12% of your sales Option B: Base salary of$20,000 a year, with a commission of 5% of your sales

How much jewelry would you need to sell for option A to produce a larger income?

12. When hired at a new job selling electronics, you are given two pay options:

Option A: Base salary of $14,000 a year, with a commission of 10% of your sales Option B: Base salary of$19,000 a year, with a commission of 4% of your sales

How much electronics would you need to sell for option A to produce a larger income?

13. Find the area of a triangle bounded by the $$y$$ axis, the line $$f(x)=9-\dfrac{6}{7} x$$, and the line perpendicular to $$f(x)$$ that passes through the origin.

14. Find the area of a triangle bounded by the $$x$$ axis, the line $$f(x) =12-\dfrac{1}{3} x$$, and the line perpendicular to $$f(x)$$ that passes through the origin.

15. Find the area of a parallelogram bounded by the $$y$$ axis, the line $$x=3$$, the line $$f(x)=1+2x$$, and the line parallel to $$f(x)$$ passing through (2, 7)

16. Find the area of a parallelogram bounded by the $$x$$ axis, the line $$g(x)=2$$, the line $$f(x)=3x$$, and the line parallel to $$f(x)$$ passing through (6, 1)

17. If $$b>0$$ and $$m<0$$, then the line $$f(x)=b+mx$$ cuts off a triangle from the first quadrant. Express the area of that triangle in terms of m and b. [UW]

18. Find the value of m so the lines $$f(x)=mx+5$$ and $$g(x)=x$$ and the $$y$$-axis form a triangle with an area of 10. [UW]

19. The median home values in Mississippi and Hawaii (adjusted for inflation) are shown below. If we assume that the house values are changing linearly,

 Year Mississippi Hawaii 1950 25200 74400 2000 71400 272700

a. In which state have home values increased at a higher rate?
b. If these trends were to continue, what would be the median home value in Mississippi in 2010?
c. If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd)

20. The median home value ins Indiana and Alabama (adjusted for inflation) are shown below. If we assume that the house values are changing linearly,

 Year Indiana Alabama 1950 37700 27100 2000 94300 85100

a. In which state have home values increased at a higher rate?
b. If these trends were to continue, what would be the median home value in Indiana in 2010?
c. If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd)

21. Pam is taking a train from the town of Rome to the town of Florence. Rome is located 30 miles due West of the town of Paris. Florence is 25 miles East, and 45 miles North of Rome. On her trip, how close does Pam get to Paris? [UW]

22. You’re flying from Joint Base Lewis-McChord (JBLM) to an undisclosed location 226 km south and 230 km east. Mt. Rainier is located approximately 56 km east and 40 km south of JBLM. If you are flying at a constant speed of 800 km/hr, how long after you depart JBLM will you be the closest to Mt. Rainier?

Answer

1a. 696 people
b. 4 years
c. 174 people per year
d. 305 people
e. $$P(t) = 305 + 174t$$
f. 2219 people.

3a. $$C(x) = 0.15x + 10$$
b. The flat monthly fee is $10 and there is an additional$0.15 fee for each additional minute used
c. $113.05 5a. $$P(t) = 190t + 4170$$ b. 6640 moose 7a. $$R(t) = 16 - 2.1t$$ b. 5.5 billion cubic feet c. During the year 2017 9. More than 133 minutes 11. More than$42857.14 worth of jewelry

13. 20.012 square units

15. 6 square units

17. $$A = - \dfrac{b^2}{2m}$$

19a. Hawaii
b. \$80640
c. During the year 1933

21. 26.225 miles