A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

The dimensions that maximize the area of the window are:

Width: 8.4 ft

Height: 4.2 ft

Step-by-step explanation:

We have a Norman window, in which the diameter of the semicircle is equal to the width of the rectangle.

The perimeter is made of 2 heights of the rectangle, one width of the rectangle plus half a circumference.

Then, if w is the diameter of the semicircle (and the width of the rectangle) and h is the other side of the rectangle we can write the perimeter as:

With the restriction for the perimeter (P=30), we can express h in funtion of w.

Now, we can express the area only in function of w and maximize its value:

To maximize the area, we derive and equal to 0:

Now, we can calculate the height as:

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