Math Help - Optimization problem

1. Optimization problem

HIii, i have a math problem

A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, wheras the semicircle is of tinted glass and only transmits half the amount of light as the clear glass. The total perimeter is fixed. Find the proportions of the window that will admit the most light.

This is what i know so far

The overall perimeter will be perimeter of rectangle plus circumference of the semi-circle. P = (Pi)(R) + 4(Pi)+ 2H

Now we need to find area which will be area of half circle plus area of rectangle.

A = {(Pi)(r)^2}/4 + 2(Pi)(H)

now i know that we have to isolate for H in the perimeter formula and sub it into the area formula....but no numbers are given..how do i find proportions?? do i just sub in any number for P??

2. Ok, here's my shot at this:

First you have done well so far, except that the formula will give you:

A = {(Pi)(r)^2}/4 + 2(r)(H) ... instead of : A = {(Pi)(r)^2}/4 + 2(Pi)(H)

Now, use the fact that:

Perimeter = pi*r + 4r + 2H , solve for H:

H = [P - pi*r - 4r]/2 ; where P is the Perimeter, which you were told it fixed; thus being a constant and (P)' = 0

Now you can follow the usual optimization process.

3. Originally Posted by umeris
HIii, i have a math problem

A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, wheras the semicircle is of tinted glass and only transmits half the amount of light as the clear glass. The total perimeter is fixed. Find the proportions of the window that will admit the most light.

This is what i know so far

The overall perimeter will be perimeter of rectangle plus circumference of the semi-circle. P = (Pi)(R) + 4(Pi)+ 2H
The perimeter is that of a semi circle $\pi R$, the two sides $2H$ and the base $2R$ for a total of:

$P=\pi R +2H +2 R$

CB

4. thnxxx

hey thanks a lot....that makes sense

5. hey u know how u say
P=\pi R +2H +(2 R) i believe it is 4R because its not only a base...it is a complete rectangular frame with the semi circular frame on top

6. Originally Posted by umeris
hey u know how u say
P=\pi R +2H +(2 R) i believe it is 4R because its not only a base...it is a complete rectangular frame with the semi circular frame on top
I made a mistake on my first analysis:

The perimeter is P = pi*R + 2R + 2H

Draw the picture and it'll make sense

7. Originally Posted by umeris
hey u know how u say
P=\pi R +2H +(2 R) i believe it is 4R because its not only a base...it is a complete rectangular frame with the semi circular frame on top
The perimeter of the window does not include the dividing line between the semi-circular and rectangular sections. Perimeter means the line or curve corresponding to the outside edge of the shape.

CB