a semicircle with a perimeter of window is constrained to be 6 metres but able to admit greatest amount of light

Question

Your father is in the process of building a new bungalow house for the whole family. Since you are an engineering student, he asked you to help in the designing of the house. He wants the windows in the shape of a rectangle capped by a semicircle with a perimeter of window is constrained to be 6 metres but able to admit greatest amount of light. Could you help your father to find the write dimension of the windows?

Answer 1

I have a feeling the intent of this problem was to include he circumference of the semi-circle as part of the perimeter.

 

The following graph depicts the window:

 

https://www.desmos.com/calculator/jdfm5acoyf

 

Drag the slider for the radius of the semi-circle to see the various possible sizes of the area.

 

Once you set up a formula for the Area of the window, you can use calculus (first derivative set to 0) to determine the value of the radius that maximizes the area, or since the resulting formula is a parabola (depicted as g(x) on the graph, you could put that in vertex form to locate the vertex.

Answer 2

x represents width window and diameter of semi-circle

3 - x represents height of window

A = x(3 - x) + (0.5)(0.5x)²

 

Convert to vertex form to determine maximum.