How do you maximize a window that consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches?

Question

Samantha Adkison · Accepted Answer

I found two values, for the width and height (of the rectangular part) of your window:
#80.6# in and #40.4# in

Considering your window as:

Emma Aldridge · Answer

undefined To maximize the area of the window, you need to optimize the dimensions of the rectangle and semicircle such that the perimeter is 288 inches. Let's denote the width of the rectangle as $ w $ inches and the height as $ h $ inches.

The perimeter $ P $ of the window is given by:
$$ P = 2w + \pi r + 2h $$

Given that $ P = 288 $ inches, we can express $ r $ in terms of $ w $ and $ h $ using the relationship between the radius and the width of the rectangle:
$$ r = \frac{w}{2} $$

Substituting this into the perimeter equation, we get:
$$ 288 = 2w + \pi \left(\frac{w}{2}\right) + 2h $$

To maximize the area, we need to differentiate the area equation with respect to $ w $, set the derivative equal to zero, and solve for $ w $. The resulting value of $ w $ will be the width that maximizes the area.

After finding the optimal value of $ w $, we can calculate the corresponding values of $ h $ and $ r $, then find the area of the rectangle and semicircle, and sum them to get the total area of the window.