A fish is reeled in at a rate of 4.0 feet per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out? (Round your answer to three decimal places.) 10 ft rad/sec
A fish is reeled in at a rate of 4.0 feet per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out? (Round your answer to three decimal places.) 10 ft rad/sec
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 Statement
A fish is reeled in at a rate of \(4.0\) feet per second from a point \(10\) feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of \(25\) feet of line out? (Round your answer to three decimal places.)
### Diagram Explanation
The provided figure illustrates the following details:
- A fishing pole is positioned \(10\) feet above the water level.
- A fish is being reeled in at a rate of \(4.0\) feet per second.
- The fishing line from the tip of the pole to the fish makes an angle \(\theta\) with the water surface.
- The length of the line between the tip of the pole and the fish is labelled \(25\) feet.
### Solution
Given:
- The rate at which the line is being pulled in, \( \frac{dl}{dt} = -4.0 \) feet/second (negative sign indicates the reeling in of the line).
- The vertical distance from the fishing pole to the water is \(10\) feet.
- The length of the fishing line at a specific moment is \(25\) feet.
To find: The rate at which the angle \(\theta\) between the line and the water is changing, represented as \( \frac{d\theta}{dt} \).
### Mathematical Approach
1. Given the vertical distance and the length of the line, use the Pythagorean theorem to find the horizontal distance, \( x \), between the point on the water directly below the pole and the fish:
\[ x^2 + 10^2 = 25^2 \]
Solve for \( x \).
2. Use trigonometric relationships and differentiation to find the rate of change of the angle \(\theta\).
3. Apply calculus methods to solve for the desired rate of change of the angle \(\theta\).
### Final Step
Insert the known values into the formula and solve for \(\frac{d\theta}{dt}\).
---
The output text and any further calculations can be used as educational content to illustrate the process of solving related rates problems in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0711daf6-3b19-448b-99c0-3e7fe08f4fe3%2Fc3e5c84b-ca7f-430b-82de-77564230cafe%2Fpvja37_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A fish is reeled in at a rate of \(4.0\) feet per second from a point \(10\) feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of \(25\) feet of line out? (Round your answer to three decimal places.)
### Diagram Explanation
The provided figure illustrates the following details:
- A fishing pole is positioned \(10\) feet above the water level.
- A fish is being reeled in at a rate of \(4.0\) feet per second.
- The fishing line from the tip of the pole to the fish makes an angle \(\theta\) with the water surface.
- The length of the line between the tip of the pole and the fish is labelled \(25\) feet.
### Solution
Given:
- The rate at which the line is being pulled in, \( \frac{dl}{dt} = -4.0 \) feet/second (negative sign indicates the reeling in of the line).
- The vertical distance from the fishing pole to the water is \(10\) feet.
- The length of the fishing line at a specific moment is \(25\) feet.
To find: The rate at which the angle \(\theta\) between the line and the water is changing, represented as \( \frac{d\theta}{dt} \).
### Mathematical Approach
1. Given the vertical distance and the length of the line, use the Pythagorean theorem to find the horizontal distance, \( x \), between the point on the water directly below the pole and the fish:
\[ x^2 + 10^2 = 25^2 \]
Solve for \( x \).
2. Use trigonometric relationships and differentiation to find the rate of change of the angle \(\theta\).
3. Apply calculus methods to solve for the desired rate of change of the angle \(\theta\).
### Final Step
Insert the known values into the formula and solve for \(\frac{d\theta}{dt}\).
---
The output text and any further calculations can be used as educational content to illustrate the process of solving related rates problems in calculus.
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