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13. After a recent Arlington High School basketball game, traffic was exiting the parking lot at a constant rate of 28 cars per minute. The parking lot started with 922 cars. (a) How many cars are still in the parking lot after 10 minutes? (b) Determine a formula for the number of cars, n. in the parking lot after m-minutes. (c) After 25 minutes, the rate at which the cars leave rises to 34 cars per minute. How many total minutes does it take for the parking lot to completely clear? Round to the nearest minute. Show your analysis.

13. After a recent Arlington High School basketball game, traffic was exiting the parking lot at a constant rate of 28 cars per minute. The parking lot started with 922 cars. (a) How many cars are still in the parking lot after 10 minutes? (b) Determine a formula for the number of cars, n. in the parking lot after m-minutes. (c) After 25 minutes, the rate at which the cars leave rises to 34 cars per minute. How many total minutes does it take for the parking lot to completely clear? Round to the nearest minute. Show your analysis.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Problem 13 After a recent Arlington High School basketball game, traffic was exiting the parking lot at a constant rate of 28 cars per minute. The parking lot started with 922 cars. **(a)** How many cars are still in the parking lot after 10 minutes? **(b)** Determine a formula for the number of cars, \( n \), in the parking lot after \( m \)-minutes. **(c)** After 25 minutes, the rate at which the cars leave rises to 34 cars per minute. How many total minutes does it take for the parking lot to completely clear? Round to the nearest minute. Show your analysis. --- ### Problem 14 A single pump is filling a storage container with water at a rate of 60 gallons per minute. After 30 minutes, an additional pump turns on and the container begins to fill at a total rate of 130 gallons per minute for an additional 30 minutes. The container already had 1,500 gallons of water when it began to be filled. **(a)** On the grid below, graph the amount of water the tank contains for the first 60 minutes. **Graph Explanation:** - The x-axis represents time in minutes from 0 to 60. - The y-axis represents volume in gallons, ranging from 0 to 7000 gallons. - From 0 to 30 minutes, the graph increases linearly starting at 1500 gallons at the rate of 60 gallons per minute. - From 30 to 60 minutes, the graph increases more steeply at the rate of 130 gallons per minute. **(b)** Write a piecewise defined function for the volume, \( V \), as a function of time, \( t \), measured in minutes.
Transcribed Image Text:### Problem 13 After a recent Arlington High School basketball game, traffic was exiting the parking lot at a constant rate of 28 cars per minute. The parking lot started with 922 cars. **(a)** How many cars are still in the parking lot after 10 minutes? **(b)** Determine a formula for the number of cars, \( n \), in the parking lot after \( m \)-minutes. **(c)** After 25 minutes, the rate at which the cars leave rises to 34 cars per minute. How many total minutes does it take for the parking lot to completely clear? Round to the nearest minute. Show your analysis. --- ### Problem 14 A single pump is filling a storage container with water at a rate of 60 gallons per minute. After 30 minutes, an additional pump turns on and the container begins to fill at a total rate of 130 gallons per minute for an additional 30 minutes. The container already had 1,500 gallons of water when it began to be filled. **(a)** On the grid below, graph the amount of water the tank contains for the first 60 minutes. **Graph Explanation:** - The x-axis represents time in minutes from 0 to 60. - The y-axis represents volume in gallons, ranging from 0 to 7000 gallons. - From 0 to 30 minutes, the graph increases linearly starting at 1500 gallons at the rate of 60 gallons per minute. - From 30 to 60 minutes, the graph increases more steeply at the rate of 130 gallons per minute. **(b)** Write a piecewise defined function for the volume, \( V \), as a function of time, \( t \), measured in minutes.
**Educational Content on Piecewise Functions and Intercepts** **15. Determine a piecewise equation for the function shown graphed below.** *Description of Graph:* - The graph depicts two linear segments on a coordinate plane. - The first line segment starts at the point (-5, 7) and connects to an intermediate point. - The second line segment starts from another intermediate point and ends at the point (5, -5). - The segments seem to intersect perpendicularly, but precise equations of these lines are needed by analyzing the plot's turn points. **16. For the piecewise function \( g(x) = \begin{cases} \frac{x}{3} + 2 & \text{if } x < 1 \\ 4x - 2 & \text{if } x \geq 1 \end{cases} \),** answer the following questions. Explain your thinking and show your work. (a) **What is the y-intercept of this function?** (b) **Determine the x-intercept(s) of this function.** (c) **Which has the greater average rate of change over the interval \(-12 \leq x \leq 8\), the function \( g(x) \) or the function \( f(x) = 2x + 7 \)?** (d) **Provide evidence that this function is not one-to-one. Explain how your evidence supports that \( g(x) \) is not one-to-one.** *Answer Key and Explanation:* (a) To find the y-intercept of \( g(x) \), evaluate both expressions at the point of their boundary, \( x = 1 \). The y-intercept is typically found when \( x = 0 \), but confirms by deriving from the segments around that point of discontinuity in the domain: - \( x = 0 \) within the first segment’s validity: \( g(0) = \frac{0}{3} + 2 = 2 \). (b) Solve each equation for when \( g(x) = 0 \) to find x-intercepts: - For \( x < 1 \), \( \frac{x}{3} + 2 = 0 \Rightarrow x = -6 \). - For \( x \geq 1 \), \( 4x - 2 = 0 \Rightarrow x = 0.
Transcribed Image Text:**Educational Content on Piecewise Functions and Intercepts** **15. Determine a piecewise equation for the function shown graphed below.** *Description of Graph:* - The graph depicts two linear segments on a coordinate plane. - The first line segment starts at the point (-5, 7) and connects to an intermediate point. - The second line segment starts from another intermediate point and ends at the point (5, -5). - The segments seem to intersect perpendicularly, but precise equations of these lines are needed by analyzing the plot's turn points. **16. For the piecewise function \( g(x) = \begin{cases} \frac{x}{3} + 2 & \text{if } x < 1 \\ 4x - 2 & \text{if } x \geq 1 \end{cases} \),** answer the following questions. Explain your thinking and show your work. (a) **What is the y-intercept of this function?** (b) **Determine the x-intercept(s) of this function.** (c) **Which has the greater average rate of change over the interval \(-12 \leq x \leq 8\), the function \( g(x) \) or the function \( f(x) = 2x + 7 \)?** (d) **Provide evidence that this function is not one-to-one. Explain how your evidence supports that \( g(x) \) is not one-to-one.** *Answer Key and Explanation:* (a) To find the y-intercept of \( g(x) \), evaluate both expressions at the point of their boundary, \( x = 1 \). The y-intercept is typically found when \( x = 0 \), but confirms by deriving from the segments around that point of discontinuity in the domain: - \( x = 0 \) within the first segment’s validity: \( g(0) = \frac{0}{3} + 2 = 2 \). (b) Solve each equation for when \( g(x) = 0 \) to find x-intercepts: - For \( x < 1 \), \( \frac{x}{3} + 2 = 0 \Rightarrow x = -6 \). - For \( x \geq 1 \), \( 4x - 2 = 0 \Rightarrow x = 0.
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