Question 1045811: CALCULUS(MAXIMA AND MINIMA):FIND THE MAXIMUM AREA OF A RECTANGLE INSCRIBED IN A SEMI-CIRCLE OF RADIUS 5 INCHES IF ITS BASE LIES ALONG THE DIAMETER OF THE SEMI-CIRCLE.
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website!  .
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As you can see from the diagram, the area of the rectangle is,
However there is a relationship between x and y because they lie on a circle,
So then,
To simplify the problem, I'm going to solve for the values of x and y that maximize the square of the area (I'll call that Z).
That's the same solution as the area because if the area is the maximum, then the square of that area will also be the maximum.
This simplifies the math.
So then,
I can make Z a function of one variable by substituting from the circle equation,
So now I have a function of one variable.
To find the maximum, take the derivative and set it equal to zero.
So there are three possible solutions,
 Then there is no rectangle.
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 Leads to a negative x value that doesn't make sense in this problem.
So then when,
So then the maximum area,
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