To solve this optimization problem, we need to maximize the area of the Norman window while keeping the perimeter constrained to 30 ft.

Step 1: Define the Variables

• Let $w$ be the width of the rectangle (also the diameter of the semicircle).
• Let $h$ be the height of the rectangle.

Step 2: Define the Perimeter Constraint

The total perimeter includes the perimeter of the rectangle (minus the top, replaced by the semicircle) and the semicircle's circumference: $w + 2h + \frac{\pi w}{2} = 30$

Step 3: Define the Area to Maximize

The area of the Norman window includes the rectangle's area and the semicircle's area: $A = w h + \frac{\pi w^2}{8}$

Step 4: Solve for $h$ in Terms of $w$ Using the Perimeter Constraint

From the perimeter equation: $h = \frac{30 - w - \frac{\pi w}{2}}{2}$

Step 5: Substitute $h$ into the Area Equation

Substitute $h$ into the area formula: $A = w \left(\frac{30 - w - \frac{\pi w}{2}}{2}\right) + \frac{\pi w^2}{8}$

Simplify: $A = \frac{w (30 - w - \frac{\pi w}{2})}{2} + \frac{\pi w^2}{8}$ $A = \frac{30w - w^2 - \frac{\pi w^2}{2}}{2} + \frac{\pi w^2}{8}$ Combine terms: $A = 15w - \frac{w^2}{2} - \frac{\pi w^2}{4} + \frac{\pi w^2}{8}$ $A = 15w - \frac{w^2}{2} - \frac{\pi w^2}{8}$

Step 6: Differentiate $A$ with Respect to $w$

To maximize $A$, take the derivative of $A$ with respect to $w$ and set it to 0: $\frac{dA}{dw} = 15 - w - \frac{\pi w}{4} = 0$

Solve for $w$: $15 = w + \frac{\pi w}{4}$ $15 = w \left(1 + \frac{\pi}{4}\right)$ $w = \frac{15}{1 + \frac{\pi}{4}}$ $w = \frac{60}{4 + \pi}$

Step 7: Find $h$

Substitute $w = \frac{60}{4 + \pi}$ into the equation for $h$: $h = \frac{30 - w - \frac{\pi w}{2}}{2}$

Step 8: Numerical Approximation

Using approximations for calculations ($\pi \approx 3.14$): $w \approx \frac{60}{4 + 3.14} \approx 8.24 \, \text{ft}$ $h = \frac{30 - 8.24 - \frac{3.14 \times 8.24}{2}}{2} \approx 5.59 \, \text{ft}$

Final Dimensions

• Width ($w$): approximately 8.24 ft
• Height ($h$): approximately 5.59 ft

Would you like a detailed numerical verification or further exploration?

Here are some related questions:

1. What happens to the area if the perimeter increases to 40 ft?
2. How does the result change if the semicircle is replaced by a full circle?
3. Can we derive the dimensions symbolically without numerical approximations?
4. What is the significance of $\pi$ in shaping the optimal window dimensions?
5. How would a different constraint (like area instead of perimeter) change the solution?

Tip: Always double-check derivative solutions by substituting back to confirm maxima or minima!