# find the the dimensions of x and y

• Dec 3rd 2009, 12:33 AM
skorpiox
find the the dimensions of x and y
a. Find a formula for the area
b. find a formula for the perimeter
c. find the dimensions x and y that maximize the area given that the perimeter is 100 .
* See attachment for details!!!

I know that this figure is composed of 4 semicircles and one rectangle ; the area of the rectangle is xy , and the area of the semicircle is 1/2pir^2. The perimeter of the rectangle is 2x + 2y , but I'm not sure how to apply the perimeter of the semicircles to this problem. On the other hand , the constraint I should use is in part c ; however , I need to find out a and b before that. I really would appreciate help for this application .Thanks for your time !!!
• Dec 3rd 2009, 01:43 AM
nikhil
here it is
1)area= (pi/4)[x^2+y^2]+xy
2)perimeter=p=pi(x+y)
3)for maximum area(constant perimeter)
put x=0 or y=0
hence maximum area will be p^2/(4pi)
though not asked but minimum area will be
[p^2/4(pi)^2][pi/2+1][this can be obtained using calculus]
• Dec 3rd 2009, 02:21 AM
HallsofIvy
Quote:

Originally Posted by skorpiox
a. Find a formula for the area
b. find a formula for the perimeter
c. find the dimensions x and y that maximize the area given that the perimeter is 100 .
* See attachment for details!!!

I know that this figure is composed of 4 semicircles and one rectangle ; the area of the rectangle is xy , and the area of the semicircle is 1/2pir^2. The perimeter of the rectangle is 2x + 2y , but I'm not sure how to apply the perimeter of the semicircles to this problem. On the other hand , the constraint I should use is in part c ; however , I need to find out a and b before that. I really would appreciate help for this application .Thanks for your time !!!

If you call the dimensions of the rectangle x and y, then you really have two complete circles with radii x/2 and y/2. The area of the entire figure is The area of those two circles, $\pi r^2= \pi x^2/4$ and $\pi y^2/4$ plus the area of the rectangle, xy. That is, the total area of the figure is $xy+ \pi x^2/4+ \pi y^2/4$.

The perimeter, however, has nothing to do with the perimeter of the rectangle. It is, instead, the total perimeter of the two circles, $\pi x+ \pi y= \pi (x+ y)$.

You are then given that $\pi (x+ y)= 100$ so $y= \frac{100}{\pi}- x$. Replace y by that in the formula for the area, $xy+ \pi x^2/4+ \pi y^2/4= x\left(\frac{100}{\pi}- x\right)+ \pi x^2/4+ \pi\left(\frac{100}{\pi}- x\right)/4$. Minimize that.
• Dec 3rd 2009, 09:39 AM
skorpiox
now, I have a better picture of this problem; I never thought about using two circles for getting my perimeter!!!!