A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 40ft that will admit the greatest possible light.  (Round the answers to two decimal places)

To maximize the amount of light admitted through a Norman window, we need to maximize the area of the window, given a fixed perimeter of 40 feet.

### Problem Breakdown:
1. **Structure of the Norman Window**:
   - The window consists of a rectangle topped by a semicircle.
   - Let the width of the rectangle (and diameter of the semicircle) be \( w \).
   - Let the height of the rectangle be \( h \).

2. **Perimeter Calculation**:
   - The perimeter consists of the width \( w \), the two heights \( h \), and the semicircle perimeter.
   - The semicircle's perimeter is \( \frac{1}{2} \times 2\pi \times \frac{w}{2} = \frac{\pi w}{2} \).
   - Total perimeter: \( P = w + 2h + \frac{\pi w}{2} \).

   Given \( P = 40 \) feet:
   \[
   40 = w + 2h + \frac{\pi w}{2}
   \]

   Simplifying for \( h \):
   \[
   h = \frac{40 - w - \frac{\pi w}{2}}{2}
   \]

3. **Area Calculation**:
   - The area \( A \) of the window is the sum of the rectangle's area and the semicircle's area:
   \[
   A = wh + \frac{1}{2}\pi\left(\frac{w}{2}\right)^2 = wh + \frac{\pi w^2}{8}
   \]

   Substituting \( h \) from the perimeter equation:
   \[
   A(w) = w \cdot \frac{40 - w - \frac{\pi w}{2}}{2} + \frac{\pi w^2}{8}
   \]

4. **Maximizing the Area**:
   - We maximize \( A(w) \) with respect to \( w \) by taking the derivative and setting it to zero.

Let's solve this to find the optimal dimensions \( w \) and \( h \).The dimensions of the Norman window that will admit the greatest possible light, given a perimeter of 40 feet, are:

- **Width (w):** 11.20 feet
- **Height (h):** 5.60 feet

These values are rounded to two decimal places.

Would you like more details on how the optimization was done, or do you have any other questions?

Here are 8 related questions you might find interesting:
1. How is the derivative used in optimization problems?
2. What are critical points in calculus?
3. Why does the semicircle influence the perimeter equation?
4. Can the shape of the window affect the optimization result?
5. What if the perimeter was different? How would the optimal dimensions change?
6. How do we confirm that the solution is a maximum?
7. What role does symmetry play in optimization problems?
8. Can you provide a step-by-step solution to this problem?

**Tip:** Optimization problems often require balancing multiple constraints to find an ideal solution, such as maximizing area while minimizing material use.

Discover how to maximize light admission through a Norman window with a perimeter of 40 feet. Learn the optimal dimensions of the window (width: 11.20 feet, height: 5.60 feet) rounded to two decimal places. This problem involves geometric optimization, perimeter calculation, and area maximization.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation