Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pers (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of each of two sides and three dividers. Let y denote the length of the other two sides. (c) Write an expression for the total area A in terms of both x and y. A= X (d) Use the given information to write an equation that relates the variables. y = +375. (e) Use part (d) to write the total area as a function of one variable. A(x) = x (f) Finish solving the problem by finding the largest area. 14,062 5 Submit Answer
Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pers (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of each of two sides and three dividers. Let y denote the length of the other two sides. (c) Write an expression for the total area A in terms of both x and y. A= X (d) Use the given information to write an equation that relates the variables. y = +375. (e) Use part (d) to write the total area as a function of one variable. A(x) = x (f) Finish solving the problem by finding the largest area. 14,062 5 Submit Answer
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need help with parts d and e
![### Problem Statement
**Scenario:**
A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. The question asks for the largest possible total area of the four pens.
**Tasks:**
1. **Draw several diagrams** illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Observe if there is a maximum area and estimate it if so.
2. **Draw a diagram illustrating the general situation**. Let \( x \) denote the length of each of two sides and three dividers. Let \( y \) denote the length of the other two sides.
3. **Write an expression for the total area** \( A \) in terms of both \( x \) and \( y \).
4. Use the given information to **write an equation that relates the variables**.
Given equation:
\[ y = \left( -\frac{5}{2} \right) x + 375 \]
5. Use part (d) to **write the total area** as a function of one variable.
6. **Finish solving the problem by finding the largest area**.
### Solution Steps
#### Step (a)
Draw diagrams with variations of pen dimensions:
- Shallow, wide pens.
- Deep, narrow pens.
- Calculate areas for each configuration.
#### Step (b)
Illustrate the general setup:
- Consider the total length of the fencing.
- Divide the fencing to represent two sets of lengths \( x \) and \( y \).
#### Step (c)
Write an expression for the total area \( A \):
\[ A = x \times y \]
#### Step (d)
Utilize the given relationship:
\[ y = \left( -\frac{5}{2} \right) x + 375 \]
#### Step (e)
Substitute \( y \) in the area expression:
\[ A(x) = x \left( \left( -\frac{5}{2} \right) x + 375 \right) \]
\[ A(x) = -\frac{5}{2} x^2 + 375x \]
#### Step (f)
Finding the maximum area:
- Identify the vertex of the parabola given by \( A(x) \).
- Use the formula for the vertex of \( ax^2 + bx +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8dd5bb-1368-4a10-9929-9294802d0b74%2Fa44ddcdb-b48e-4d03-ba85-b6903fb87ce1%2F1dvet0g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Scenario:**
A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. The question asks for the largest possible total area of the four pens.
**Tasks:**
1. **Draw several diagrams** illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Observe if there is a maximum area and estimate it if so.
2. **Draw a diagram illustrating the general situation**. Let \( x \) denote the length of each of two sides and three dividers. Let \( y \) denote the length of the other two sides.
3. **Write an expression for the total area** \( A \) in terms of both \( x \) and \( y \).
4. Use the given information to **write an equation that relates the variables**.
Given equation:
\[ y = \left( -\frac{5}{2} \right) x + 375 \]
5. Use part (d) to **write the total area** as a function of one variable.
6. **Finish solving the problem by finding the largest area**.
### Solution Steps
#### Step (a)
Draw diagrams with variations of pen dimensions:
- Shallow, wide pens.
- Deep, narrow pens.
- Calculate areas for each configuration.
#### Step (b)
Illustrate the general setup:
- Consider the total length of the fencing.
- Divide the fencing to represent two sets of lengths \( x \) and \( y \).
#### Step (c)
Write an expression for the total area \( A \):
\[ A = x \times y \]
#### Step (d)
Utilize the given relationship:
\[ y = \left( -\frac{5}{2} \right) x + 375 \]
#### Step (e)
Substitute \( y \) in the area expression:
\[ A(x) = x \left( \left( -\frac{5}{2} \right) x + 375 \right) \]
\[ A(x) = -\frac{5}{2} x^2 + 375x \]
#### Step (f)
Finding the maximum area:
- Identify the vertex of the parabola given by \( A(x) \).
- Use the formula for the vertex of \( ax^2 + bx +
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