HELP WITH A PROBLEM - FUNCTION

[sungjin6458]

well i dont get this...

Q. A special window in the shape of a rectangle with semicircles at each end is to be constructed so that the outside dimensions are 100 feet in length. Find the dimensions that maximizes the area of the rectangle.


so far i got:
Length=L
diameter=D

L+D=100
then i got stuck already from there............this problem is hard for me thx in advance

 

[galactus]

Well, let's see. If we let the rectangle be length x and width y, then the semicircles will have perimeter \(\displaystyle {\pi}\)y. So the entire perimeter can be expressed as

\(\displaystyle 2x+2y+{\pi}y=100--->2x+y(2+{\pi})=100\)

The area can be expressed as \(\displaystyle xy+2(\frac{y}{2})^{2}{\pi}\)=

\(\displaystyle xy+\frac{y^{2}}{2}{\pi}\)=A.


You can solve the first equation for y, sub it into the area equation and differentiate, set equal to 0 and solve for x. You will find the rectangle will be a square.