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Math Help - find the the dimensions of x and y

  1. #1
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    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 !!!
    Attached Thumbnails Attached Thumbnails find the the dimensions of x and y-img054.jpg  
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  2. #2
    Senior Member nikhil's Avatar
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    Lightbulb 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]
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  3. #3
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    Quote Originally Posted by skorpiox View Post
    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.
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  4. #4
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    now, I have a better picture of this problem; I never thought about using two circles for getting my perimeter!!!!
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