Hello Ife Originally Posted by

**Ife** A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only one-fourth as much light per unit area as clear glass does. The total perimeter is fixed. FInd the proportions of the window that will admit the most light. Neglect the thickness of the frame.

Now i am thinking the solution to this is something along the lines of:

$\displaystyle \frac {1}{2} \Pi D = \frac{1}{4} D x B $ (length by breath, where the length is equal to the diameter)

When this is simplified, i am getting $\displaystyle \frac{1}{4} \Pi : \frac{1}{2}B$

Is that a logical process to approach this??

Thanks 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 $\displaystyle h$, and the radius of the semi-circle is $\displaystyle r$. Then, the width of the rectangle is $\displaystyle 2r$. So the total perimeter is: $\displaystyle 2h+2r+\pi r = p$, say, where $\displaystyle p$ is a constant.

$\displaystyle \Rightarrow h = \tfrac12(p-[\pi+2]r)$ (1)

Now consider the areas. The area of the rectangle is:$\displaystyle 2rh = r(p-[\pi+2]r)$, from equation (1)

and the area of the semicircle is:$\displaystyle \tfrac12\pi r^2$

Now suppose that the glass in the semi-circle admits $\displaystyle 1$ unit of light per unit of area. Then the rectangle glass admits $\displaystyle 4$ units of light per unit area. So the total light admitted, $\displaystyle L$, is given by:$\displaystyle L = \tfrac12\pi r^2+4r(p-[\pi+2]r)$

OK. So you now need to:

- Differentiate with respect to $\displaystyle r$, and equate the result to zero

- Solve the resulting equation, to get $\displaystyle r$ in terms of $\displaystyle p$

- Check that this value of $\displaystyle r$ gives a maximum value of $\displaystyle L$

- Substitute into (1) to get $\displaystyle h$ in terms of $\displaystyle p$

- Write down and simplify the ratio $\displaystyle h:2r$ to get the proportions of the rectangle when the most light is admitted

I make the ratio $\displaystyle h:2r = (8+3\pi):16$. Do you agree?

Grandad