Of all rectangles with a perimeter of 20, which one has the maximum area? ... If x and y are the length and width of the rectangle, respectively, then the area of the rectangle is A = xy, Writing the area function as a function of x, it follows that the area is A(x) = Evaluate A at the endpoints of and at the critical point of A. It follows that A where 2x + 2y = where ≤x≤ has an absolute maximum value at x = length of and a width of (Simplify your answers.) . Therefore, the rectangle that has the maximum area has a
Of all rectangles with a perimeter of 20, which one has the maximum area? ... If x and y are the length and width of the rectangle, respectively, then the area of the rectangle is A = xy, Writing the area function as a function of x, it follows that the area is A(x) = Evaluate A at the endpoints of and at the critical point of A. It follows that A where 2x + 2y = where ≤x≤ has an absolute maximum value at x = length of and a width of (Simplify your answers.) . Therefore, the rectangle that has the maximum area has a
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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Of all rectangles with a perimeter of 20, which one has the maximum area?
If x and y are the length and width of the rectangle, respectively, then the area of the rectangle is
A=xy,
where
2x+2y=enter your response here.
Writing the area function as a function of x, it follows that the area is
A(x)=enter your response here,
where
enter your response here≤x≤enter your response here.
Evaluate A at the endpoints of
enter your response here,enter your response here
and at the critical point of A. It follows that A has an absolute maximum value at
x=enter your response here.
Therefore, the rectangle that has the maximum area has a length of
enter your response here
and a width of
enter your response here.
![**Title: Maximizing the Area of a Rectangle with a Given Perimeter**
**Problem Statement:**
Of all rectangles with a perimeter of 20, which one has the maximum area?
**Solution Approach:**
1. **Defining the Variables:**
- Let \( x \) and \( y \) represent the length and width of the rectangle, respectively.
- The area \( A \) of the rectangle is given by:
\[
A = xy
\]
2. **Perimeter Condition:**
- The perimeter of the rectangle is 20, so:
\[
2x + 2y = 20
\]
- Simplifying, we get:
\[
2x + 2y = 20 \implies 2x + 2y = 10
\]
3. **Expressing the Area in Terms of \( x \):**
- Solve for \( y \):
\[
y = \frac{10 - 2x}{2}
\]
- Substitute \( y \) in the area equation:
\[
A(x) = x \left(\frac{10 - 2x}{2}\right)
\]
4. **Solving for Maximum Area:**
- Determine the domain of \( x \):
\[
0 \leq x \leq 5
\]
- Evaluate \( A(x) \) at the endpoints and the critical point obtained from differentiation:
\[
\left(\frac{10 - 2x}{2}\right)
\]
\[
A(x) = -2x^2 + 10x
\]
5. **Conclusion:**
- The absolute maximum value occurs at \( x = 5 \).
- Therefore, the rectangle with maximum area has dimensions such that its length \( = 5 \) and width \( = 5 \).
(Simplify your answers where necessary.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffef37ab4-92ab-41e4-8960-935917e1958c%2F287be336-0a33-414c-b4b3-c0f37fbfc81e%2Ftqbu3v9_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Maximizing the Area of a Rectangle with a Given Perimeter**
**Problem Statement:**
Of all rectangles with a perimeter of 20, which one has the maximum area?
**Solution Approach:**
1. **Defining the Variables:**
- Let \( x \) and \( y \) represent the length and width of the rectangle, respectively.
- The area \( A \) of the rectangle is given by:
\[
A = xy
\]
2. **Perimeter Condition:**
- The perimeter of the rectangle is 20, so:
\[
2x + 2y = 20
\]
- Simplifying, we get:
\[
2x + 2y = 20 \implies 2x + 2y = 10
\]
3. **Expressing the Area in Terms of \( x \):**
- Solve for \( y \):
\[
y = \frac{10 - 2x}{2}
\]
- Substitute \( y \) in the area equation:
\[
A(x) = x \left(\frac{10 - 2x}{2}\right)
\]
4. **Solving for Maximum Area:**
- Determine the domain of \( x \):
\[
0 \leq x \leq 5
\]
- Evaluate \( A(x) \) at the endpoints and the critical point obtained from differentiation:
\[
\left(\frac{10 - 2x}{2}\right)
\]
\[
A(x) = -2x^2 + 10x
\]
5. **Conclusion:**
- The absolute maximum value occurs at \( x = 5 \).
- Therefore, the rectangle with maximum area has dimensions such that its length \( = 5 \) and width \( = 5 \).
(Simplify your answers where necessary.)
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