Farmer John has 1200 feet of fencing and wishes to use it to fence a rectangular plot divided into two subplots as in the figure below. Suppose we want to determine the dimensions w and h of the plot with maximum area. h If Farmer John uses all the fencing, then 3h + 2w 1200 and so the area of the region is Α(w) - υ 1200–2w 3 Determine the values of w and h that give the maximum area. The exact value of w s... The exact value of h is...
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Answer both questions for the values of w and h
![**Optimization Problem: Maximizing Area**
Farmer John has 1200 feet of fencing and wishes to use it to fence a rectangular plot divided into two subplots as in the figure below. Suppose we want to determine the dimensions \( w \) and \( h \) of the plot with maximum area.
![Rectangular plot divided into two subplots](image link here, if any)
If Farmer John uses all the fencing, then \( 3h + 2w = 1200 \) and so the area of the region is
\[ A(w) = w \left( \frac{1200 - 2w}{3} \right) \]
Determine the values of \( w \) and \( h \) that give the maximum area.
---
**The exact value of \( w \) is:**
[Input box for \( w \) value]
---
**The exact value of \( h \) is:**
[Input box for \( h \) value]
---
### Diagram Explanation:
- The diagram shows a large rectangle divided into two equal smaller rectangular sections by a vertical line.
- The total width of the large rectangle is denoted by \( w \).
- The height of the large rectangle is denoted by \( h \).
- The perimeter constraint is given by the equation \( 3h + 2w = 1200 \).
### Steps to Solve the Problem:
1. **Setting up the Perimeter Equation:**
Since the total fencing used around the border and the dividing line is 1200 feet, we have the equation \( 3h + 2w = 1200 \).
2. **Express \( h \) in terms of \( w \):**
\[
h = \frac{1200 - 2w}{3}
\]
3. **Formulate the Area Function:**
Substituting \( h \) in terms of \( w \) into the area formula \( A = w \cdot h \):
\[
A(w) = w \left(\frac{1200 - 2w}{3} \right) = \frac{1200w - 2w^2}{3}
\]
4. **Maximizing the Area:**
Find the value of \( w \) that maximizes \( A(w) \). This involves differentiating \( A(w) \) with respect to \( w \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c9fe6be-99fb-4863-a04f-7bda979173e6%2Fb960e714-0b55-478b-80ad-bb5132109d7c%2F0j8y91q_processed.png&w=3840&q=75)
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