Hello IfeThanks for showing us your thinking. But you're approaching this in the wrong way. You need to set up some variables to represent the height and radius, and then represent the conditions in the question using these variables. I'll start you off.

Suppose that the height of the rectangle is , and the radius of the semi-circle is . Then, the width of the rectangle is . So the total perimeter is:, say, where is a constant.Now consider the areas. The area of the rectangle is:

(1)

, from equation (1)and the area of the semicircle is:

Now suppose that the glass in the semi-circle admits unit of light per unit of area. Then the rectangle glass admits units of light per unit area. So the total light admitted, , is given by:

OK. So you now need to:

- Simplify this expression

- Differentiate with respect to , and equate the result to zero

- Solve the resulting equation, to get in terms of

- Check that this value of gives a maximum value of

- Substitute into (1) to get in terms of

- Write down and simplify the ratio to get the proportions of the rectangle when the most light is admitted

I make the ratio . Do you agree?

Grandad