To solve the given problem involving the Norman window, let's break it down step-by-step.

1. Understanding the components:

• The window consists of:
  1. A semicircle at the top with radius $r$.
  2. A rectangle below the semicircle, with:
     • Width $2r$ (since the width of the rectangle equals the diameter of the semicircle).
     • Height $w$ (denoted in the figure).

2. Perimeter constraint:

The perimeter of the Norman window includes:

• The semicircle's curved edge: $\pi r$ (half of the circumference of a full circle).
• The two vertical sides of the rectangle: $2w$.
• The bottom side of the rectangle: $2r$ (same as the diameter of the semicircle).

Thus, the total perimeter is: $\pi r + 2w + 2r = 23 \, \text{ft}.$

3. Optimization goal:

We need to maximize the area to allow the most light. The total area consists of:

• Area of the semicircle: $\frac{1}{2} \pi r^2$.
• Area of the rectangle: $2r \cdot w$.

Thus, the total area is: $A(r, w) = \frac{1}{2} \pi r^2 + 2r w.$

4. Solve for $w$ in terms of $r$ using the perimeter:

From the perimeter equation: $\pi r + 2w + 2r = 23.$ Rearrange to solve for $w$: $2w = 23 - \pi r - 2r,$ $w = \frac{23 - \pi r - 2r}{2}.$

5. Express the area in terms of $r$ only:

Substitute the expression for $w$ into the area formula: $A(r) = \frac{1}{2} \pi r^2 + 2r \left( \frac{23 - \pi r - 2r}{2} \right).$

Simplify: $A(r) = \frac{1}{2} \pi r^2 + r(23 - \pi r - 2r).$ $A(r) = \frac{1}{2} \pi r^2 + 23r - \pi r^2 - 2r^2.$ $A(r) = -\frac{1}{2} \pi r^2 - 2r^2 + 23r.$

6. Maximize the area:

To find the value of $r$ that maximizes the area, take the derivative of $A(r)$ with respect to $r$ and set it equal to zero: $\frac{dA}{dr} = -\pi r - 4r + 23 = 0.$ $-(\pi + 4)r + 23 = 0.$ $(\pi + 4)r = 23.$ $r = \frac{23}{\pi + 4}.$

7. Calculate the corresponding $w$:

Now that we have $r$, substitute it into the equation for $w$: $w = \frac{23 - \pi r - 2r}{2}.$

Using $r = \frac{23}{\pi + 4}$, you can substitute this value into the expression to compute $w$.

Would you like me to proceed with the numerical calculation for these values or clarify any step?

5 Related Questions:

1. How do we ensure this critical point yields the maximum area?
2. What is the area of the Norman window with these dimensions?
3. How does the perimeter constraint limit the design?
4. What happens if the perimeter was increased or decreased?
5. Can calculus guarantee the most optimal dimensions in all cases?

Tip:

Always verify your derivative calculations to ensure you correctly identify critical points for optimization problems.