2.2.2: Exercise M.2

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(1) Just like smartphones, most car values decrease over time. For example, in 2015, a new Toyota Prius Two sold for $25,500.¹⁵ In 2023, online car valuation sites like the Kelley Blue Book and Car Gurus list used Prius Twos for $18,500. The value of the 2015 Toyota Prius Two depreciated over time. Use this information to answer the following problems.

(a) Assume the depreciation is linear. What is the rate of change of a linear model that represents the value of the Prius over time?

(b) Create a linear model of the value of the Prius, where V is the value, and t is the number of years after 2015.

(d) Use the model to estimate the value of the Prius in 2026.

(2) You have just discovered that your bathtub faucet has a leak. You don’t have the funds to get this fixed right away, so you decide to use water that leaks out to water your houseplants.

(a) Assume that you left a large measuring cup under the leaking faucet from 10:00 p.m. until 6:00 a.m. the next morning. In the morning, you find the measuring cup filled to the 3-cup mark. Now assume you put a large pot under the faucet. Fill in the table below to show the total amount of water you would find in the pot if you took measurements every two hours over a 24-hour period.

Hours	Amount of Water (Cups)
0
2
4
6
8
10
12
14
16
18
20
22
24

On a separate piece of paper, or using a spreadsheet application, make a graph of the amount of water that leaked from the faucet over time by plotting these points. What do you notice about the graph? Note: If completing this problem online, follow the instructions given online to create your graph.

(b) What is the significance of the vertical and horizontal intercepts? Explain the meaning in the context of this situation.

(c) If you used a 1-gallon watering can, how long would it take to fill up? Round to the nearest hundredth of an hour. (Hint: You may need to look up how many cups are in a gallon.)

(d) What is the rate of change of the corresponding linear model? In other words, how fast is the water leaking from the faucet?

(e) Write an equation modeling the quantity of water that has leaked from the faucet over time. Be sure to define the variables you use, including units.

(f) Using your model, how much water would leak out in 11 hours and 15 minutes? Round your answer to the nearest hundredth of a cup.

(3) Obesity among adults in the United States is increasing. We want to investigate how the percentage of U.S. adults who are obese is growing. The table below shows the percentage of U.S. adults who were obese (based on BMI) for each year given.¹⁶

Year	Percentage of U.S. Adults Who are Considered Obese
2011	34.0
2012	34.7
2013	35.4
2014	36.0
2015	36.7
2016	37.3

When we examine the data above, we see that the percentage of adults who are obese generally increases from year to year. However, the amount of the increase is not exactly the same every year. We will treat the data as linear for this problem, even though the data is only roughly linear. Use a spreadsheet to answer the following questions.

(a) What is the trendline for the data? (Hint: First create a scatterplot.) Adjust the year to be the number of years after 2011.

(b) What is the slope of the linear model (trendline)? What does the slope mean?

(c) Using the trendline, predict the percentage of U.S. adults that will be considered obese in 2029. Round to the nearest percent.

(d) Using your model, what percentage of U.S. adults will be considered obese in the year 2150? Is this reliable?

__________________________________________________

¹⁵ https://www.kbb.com/toyota/prius/2015/

¹⁶ https://ourworldindata.org/grapher/share-of-adults-defined-as-obese?tab=chart&country=~USA

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