A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 47 feet?

it says incorrect

p = W + 2L + πW/2 = 47

2L = 47 - W - πW/2
L = 23.5 - 1.28W

A = WL + π(W/2)^2/2
= W(23.5 - 1.28W) + .327W^2
= 23.5W - 1.28W^2 + .327W^2
= 23.5W - .953W^2

max area at W = 23.5/1.906 = 12.3 ft
L = 7.756

perimeter = 12.3 + 2*7.756 + 1.57*12.3 = 47.1 (close enough)

area = 12.3*7.756 + 3.14*6.15 = 114.7 ft^2

To find the area of the largest possible Norman window with a perimeter of 47 feet, we need to determine the dimensions of the window that maximize its area.

Let's start by defining the variables:
Let W be the width of the rectangle.
Let H be the height of the rectangle.
Let r be the radius of the semicircle.

We know that the perimeter of the window is equal to the sum of the lengths of all its sides, which can be calculated as follows:
Perimeter = 2W + H + πr + W
Given that the perimeter is 47 feet, we can write the equation:
47 = 2W + H + πr + W

Now, we need to express the area of the window in terms of W and r. The total area consists of the area of the rectangle and the area of the semicircle. Therefore, the area is given by:
Area = (W * H) + (π * r^2 / 2)

To maximize the area, we can use calculus by taking partial derivatives. However, in this case, we can simplify the problem using some geometrical observations.

Observation 1: The diameter of the semicircle is equal to the width of the rectangle (2r = W).
Observation 2: The height of the rectangle is equal to the radius of the semicircle (H = r).

Using these observations, we can rewrite the perimeter equation as:
47 = 2(2r) + r + πr + (2r)
47 = 4r + r + πr + 2r
47 = 7r + πr
47 = (7 + π)r

We can solve this equation to find the value of r:
r = 47 / (7 + π)

Now, substitute r back into the equation for the width:
W = 2r = 2 * (47 / (7 + π))

Finally, plug in the values of W and r into the area equation to calculate the area of the largest possible Norman window:
Area = (W * H) + (π * r^2 / 2) = (2 * (47 / (7 + π)) * (47 / (7 + π))) + (π * (47 / (7 + π))^2 / 2)

Using this equation, you can calculate the area of the largest possible Norman window.