A farmer wants to fence a rectangular area of 392 square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river. Length: Number feet Width: Number feet
A farmer wants to fence a rectangular area of 392 square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river. Length: Number feet Width: Number feet
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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A farmer wants to fence a rectangular area of 392 square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river.
![### Problem Statement
A farmer wants to fence a rectangular area of 392 square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river.
### Inputs
- **Length:**
- **Type:** Number
- **Unit:** feet
- **Width:**
- **Type:** Number
- **Unit:** feet
### Explanation
To minimize the amount of fencing required:
1. Let \( L \) be the length along the river and \( W \) be the width perpendicular to the river.
2. The area (\( A \)) is given by the formula \( L \times W = 392 \) square feet.
3. Since no fencing is required along the river, the total length of fencing required will be \( L + 2W \).
### Calculation Steps
1. Express \( L \) in terms of \( W \) using the area constraint:
\[
L = \frac{392}{W}
\]
2. Substitute this into the fencing equation to minimize the function:
\[
F = \frac{392}{W} + 2W
\]
3. Differentiate \( F \) with respect to \( W \) to find the critical points:
\[
F'(W) = -\frac{392}{W^2} + 2
\]
4. Solve for \( W \) when \( F'(W) = 0 \):
\[
-\frac{392}{W^2} + 2 = 0
\]
\[
\frac{392}{W^2} = 2
\]
\[
W^2 = \frac{392}{2} = 196
\]
\[
W = \sqrt{196} = 14
\]
5. Substitute \( W = 14 \) back to find \( L \):
\[
L = \frac{392}{14} = 28
\]
### Conclusion
To minimize the amount of fencing required:
- **Length:** 28 feet
- **Width:** 14 feet
This results in the minimum fencing needed while maintaining the total area of 392 square feet.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcecc0901-ec5b-491a-8e4c-23ad559e3c3b%2Fc4534ae2-18c9-473a-82ed-1ca141dfb1ee%2Fgz06ny_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A farmer wants to fence a rectangular area of 392 square feet next to a river. Find the length and width of the rectangle which uses the least amount of fencing if no fencing is needed along the river. Assume the length of the fence runs parallel to the river.
### Inputs
- **Length:**
- **Type:** Number
- **Unit:** feet
- **Width:**
- **Type:** Number
- **Unit:** feet
### Explanation
To minimize the amount of fencing required:
1. Let \( L \) be the length along the river and \( W \) be the width perpendicular to the river.
2. The area (\( A \)) is given by the formula \( L \times W = 392 \) square feet.
3. Since no fencing is required along the river, the total length of fencing required will be \( L + 2W \).
### Calculation Steps
1. Express \( L \) in terms of \( W \) using the area constraint:
\[
L = \frac{392}{W}
\]
2. Substitute this into the fencing equation to minimize the function:
\[
F = \frac{392}{W} + 2W
\]
3. Differentiate \( F \) with respect to \( W \) to find the critical points:
\[
F'(W) = -\frac{392}{W^2} + 2
\]
4. Solve for \( W \) when \( F'(W) = 0 \):
\[
-\frac{392}{W^2} + 2 = 0
\]
\[
\frac{392}{W^2} = 2
\]
\[
W^2 = \frac{392}{2} = 196
\]
\[
W = \sqrt{196} = 14
\]
5. Substitute \( W = 14 \) back to find \( L \):
\[
L = \frac{392}{14} = 28
\]
### Conclusion
To minimize the amount of fencing required:
- **Length:** 28 feet
- **Width:** 14 feet
This results in the minimum fencing needed while maintaining the total area of 392 square feet.
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