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.)

Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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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
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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