SOLUTION: A rectangle is inscribed inside the semi-circle y=square root of 100-x^2. what are the dimensions of the rectangle with maximum area? include a diagram

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Question 322321: A rectangle is inscribed inside the semi-circle y=square root of 100-x^2. what are the dimensions of the rectangle with maximum area? include a diagram
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
For a rectangle,
A=L%2AW
In this case, L=2x and W=y
A=2xy
y=sqrt%28100-x%5E2%29
A=2x%2Asqrt%28100-x%5E2%29
Now area is only a function of x.
Differentiate with respect to x to find the maximum.
dA%2Fdx=2x%2A%281%2F2%29%28100-x%5E2%29%5E%28-1%2F2%29%28-2x%29%2Bsqrt%28100-x%5E2%29%282%29
dA%2Fdx=2sqrt%28100-x%5E2%29-2x%5E2%2Fsqrt%28100-x%5E2%29
Set the derivative equal to zero.
2sqrt%28100-x%5E2%29-2x%5E2%2Fsqrt%28100-x%5E2%29=0
2sqrt%28100-x%5E2%29=2x%5E2%2Fsqrt%28100-x%5E2%29
100-x%5E2=x%5E2
2x%5E2=100
x%5E2=50
x=sqrt%2850%29=5sqrt%282%29
y=sqrt%28100-50%29=5sqrt%282%29
A=2%285sqrt%282%29%29%2A5sqrt%282%29
A=100
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