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Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which water is flowing from the tank after 15 minutes.) t (min) 5 10 15 20 25 30 111 28 0 V (gal) 694 650- 600+ 550+ 500- 450+ change in Volume change in time slope -300 400- 350+ 300+ 250+ î 9 200+ 150+ = -33. 3 100- 50+ 0 V (gallons) 57 300 approximate graph of function approximate tangent line. 9 10 15 444 250 20 25 7 (minutes) 30 = 01:58 Сс

Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which water is flowing from the tank after 15 minutes.) t (min) 5 10 15 20 25 30 111 28 0 V (gal) 694 650- 600+ 550+ 500- 450+ change in Volume change in time slope -300 400- 350+ 300+ 250+ î 9 200+ 150+ = -33. 3 100- 50+ 0 V (gallons) 57 300 approximate graph of function approximate tangent line. 9 10 15 444 250 20 25 7 (minutes) 30 = 01:58 Сс

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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### Estimating the Slope of a Tangent Line #### Objective: Use a graph of the function to estimate the slope of the tangent line at point \( P \). This slope represents the rate at which water is flowing from the tank after 15 minutes. #### Data Table: | \( t \) (min) | 5 | 10 | 15 | 20 | 25 | 30 | |---------------|-----|-----|-----|-----|-----|-----| | \( V \) (gal) | 694 | 444 | 250 | 111 | 28 | 0 | #### Graph Explanation: An approximate graph of the function is shown with time (in minutes) on the x-axis and volume (in gallons) on the y-axis. At point \( P \), which corresponds to \( t = 15 \) minutes, the function's graph and the approximate tangent line are drawn. To find the slope of the tangent line at \( P \): 1. **Identify coordinates on the tangent line:** - At \( t \approx 9 \) minutes, \( V \approx 300 \) gallons. - At \( t = 18 \) minutes, \( V \approx 0 \) gallons. 2. **Calculate the slope:** - Slope \( = \frac{\text{change in Volume}}{\text{change in time}} \) - Change in Volume \( = 0 - 300 = -300 \) gallons - Change in time \( = 18 - 9 = 9 \) minutes - Hence, slope \( \approx \frac{-300}{9} = -33.3 \) This slope indicates that the water is flowing out of the tank at a rate of approximately 33.3 gallons per minute at \( t = 15 \) minutes. Below the graph, a video screen shows a playback time of 1:57, with controls to play and pause. There is also a link provided to view the transcript of the video.
Transcribed Image Text:### Estimating the Slope of a Tangent Line #### Objective: Use a graph of the function to estimate the slope of the tangent line at point \( P \). This slope represents the rate at which water is flowing from the tank after 15 minutes. #### Data Table: | \( t \) (min) | 5 | 10 | 15 | 20 | 25 | 30 | |---------------|-----|-----|-----|-----|-----|-----| | \( V \) (gal) | 694 | 444 | 250 | 111 | 28 | 0 | #### Graph Explanation: An approximate graph of the function is shown with time (in minutes) on the x-axis and volume (in gallons) on the y-axis. At point \( P \), which corresponds to \( t = 15 \) minutes, the function's graph and the approximate tangent line are drawn. To find the slope of the tangent line at \( P \): 1. **Identify coordinates on the tangent line:** - At \( t \approx 9 \) minutes, \( V \approx 300 \) gallons. - At \( t = 18 \) minutes, \( V \approx 0 \) gallons. 2. **Calculate the slope:** - Slope \( = \frac{\text{change in Volume}}{\text{change in time}} \) - Change in Volume \( = 0 - 300 = -300 \) gallons - Change in time \( = 18 - 9 = 9 \) minutes - Hence, slope \( \approx \frac{-300}{9} = -33.3 \) This slope indicates that the water is flowing out of the tank at a rate of approximately 33.3 gallons per minute at \( t = 15 \) minutes. Below the graph, a video screen shows a playback time of 1:57, with controls to play and pause. There is also a link provided to view the transcript of the video.
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