Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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![### Norman Window Optimization Problem
A Norman window has the shape of a rectangle surmounted by a semicircle. The exterior perimeter of the window shown in the figure is 48 feet. Find the height \( h \) and the radius \( r \) that will give the maximum area.
**Hint:**
- Find a linear equation that relates \( r \) and \( h \).
- Then, using this equation, write the area of the window as a quadratic function of the radius \( r \).
#### Image Description:
The image displays a Norman window consisting of a rectangle and a semicircle on top of it. The variables \( r \) (radius) and \( h \) (height) are indicated within the diagram, with \( h \) referring to the height of the rectangle and \( r \) to the radius of the semicircle.
#### Steps to Solve the Problem:
1. **Identify the Perimeter Equation:**
- The total perimeter, \( P \), is given as 48 feet.
- The perimeter of the rectangular part includes the two heights \( h \) and the width \( 2r \) (which is the diameter of the semicircle).
- The perimeter of the semicircular part is half the circumference of a full circle, which is \( \pi r \).
The combined perimeter equation is:
\[ P = 2h + 2r + \pi r = 48 \]
2. **Relate \( h \) and \( r \):**
- Solve the perimeter equation for height \( h \):
\[ 2h + 2r + \pi r = 48 \]
\[ 2h + r(2 + \pi) = 48 \]
\[ 2h = 48 - r(2 + \pi) \]
\[ h = \frac{48 - r(2 + \pi)}{2} \]
3. **Write the Area Equation:**
- The area of the rectangle is \( A_{rectangle} = 2r \cdot h \).
- The area of the semicircle is half of the area of a full circle, \( A_{semicircle} = \frac{1}{2} \pi r^2 \).
- The total area \( A \) is:
\[ A = A_{rectangle} + A_{semic](https://content.bartleby.com/qna-images/question/f06f185d-4ef2-4eb1-9d7e-74b242cf1b9c/6a1dd959-d578-415c-b103-6e0fa70a1e48/l4f9vb9_processed.jpeg)
Transcribed Image Text:### Norman Window Optimization Problem
A Norman window has the shape of a rectangle surmounted by a semicircle. The exterior perimeter of the window shown in the figure is 48 feet. Find the height \( h \) and the radius \( r \) that will give the maximum area.
**Hint:**
- Find a linear equation that relates \( r \) and \( h \).
- Then, using this equation, write the area of the window as a quadratic function of the radius \( r \).
#### Image Description:
The image displays a Norman window consisting of a rectangle and a semicircle on top of it. The variables \( r \) (radius) and \( h \) (height) are indicated within the diagram, with \( h \) referring to the height of the rectangle and \( r \) to the radius of the semicircle.
#### Steps to Solve the Problem:
1. **Identify the Perimeter Equation:**
- The total perimeter, \( P \), is given as 48 feet.
- The perimeter of the rectangular part includes the two heights \( h \) and the width \( 2r \) (which is the diameter of the semicircle).
- The perimeter of the semicircular part is half the circumference of a full circle, which is \( \pi r \).
The combined perimeter equation is:
\[ P = 2h + 2r + \pi r = 48 \]
2. **Relate \( h \) and \( r \):**
- Solve the perimeter equation for height \( h \):
\[ 2h + 2r + \pi r = 48 \]
\[ 2h + r(2 + \pi) = 48 \]
\[ 2h = 48 - r(2 + \pi) \]
\[ h = \frac{48 - r(2 + \pi)}{2} \]
3. **Write the Area Equation:**
- The area of the rectangle is \( A_{rectangle} = 2r \cdot h \).
- The area of the semicircle is half of the area of a full circle, \( A_{semicircle} = \frac{1}{2} \pi r^2 \).
- The total area \( A \) is:
\[ A = A_{rectangle} + A_{semic
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