Question

Could someone answer this please.. I am stumped.
A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions w x h of the Norman window whose perimeter is 600in that has maximal area.

Could i get some help finding the height please?
the picture will be in the reply section

Answer 1

Let me find some paper...

Answer 2

Let me give it to you as i go along...

Answer 3

So let P stand for perimeter

P = 600 = h + h + w + w*pi/2
600 = 2h + w(1+pi/2)

from this after a bit of tidying up, w = (1200 - 4h) / (2 + pi)

Answer 4

Now let A stand for Area (which we want to maximise)

A = wh + [pi*(w/2)^2] / 2
A = wh + w^2 * pi / 8

Answer 5

This gets a bit messy (just saying)

Answer 6

i am really tired, so this is already hard to understand :)

Answer 7

Well it gets worse

Answer 8

Can i just give you the gist of it?

Answer 9

I also cannot be bothered typing in the rest

Answer 10

that sounds like a good idea. thank you.

Answer 11

So from the first bit, I got an expression for w in terms of h.
We now have A, but in terms of w and h.
We want to maximise A in terms of h, so we substitute our h's in A.
Now we have an expression for A in terms of ONLY h.

Great!

Now after heaps of cleaning up, we get that A is a concave down quadratic in h. From this we know that there must be a maximum point, and we find this by using the formula:

x = -b/(2a) in the quadratic 0 = ax^2 + bx + c

and there you have it. This x value will be your height that causes the maximum area

Answer 12

wow... that is amazing that your brain can work like that. thank you

Answer 13

but what does b stand for?

Answer 14

Oh ax^2 + bx + c is the general expression for a quadratic.
a, b and c are any real values (positive or NEGATIVE), which in your case A will be neg

Answer 15

but remember for the formula to work, ax^2 + bx + c must equal 0

Answer 16

okay thank you :)

Answer 17

No worries :D