Question 1137441
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Allow me to use a different single variable; the suggested ones are not pleasant.<br>
Let the base of the rectangle be 2x.  Obviously the center of the semicircle is the midpoint of the base of the rectangle.<br>
Then the height of the rectangle is {{{sqrt(r^2-x^2)}}}.<br>
The the area of the rectangle is base times height: {{{A = (2x)*(sqrt(r^2-x^2))}}}<br>
To maximize the area, we find the derivative of the area function using the product rule....<br>
{{{dA/dx = 2(sqrt(r^2-x^2))+2x(1/2)(1/sqrt(r^2-x^2))(-2x) = 2(sqrt(r^2-x^2))-2x^2/sqrt(r^2-x^2) = (2(r^2-x^2)-2x^2)/sqrt(r^2-x^2) = (2r^2-4x^2)/sqrt(r^2-x^2)}}}<br>
...and find where the derivative is 0 (where the numerator is 0):<br>
{{{2r^2-4x^2 = 0}}}
{{{r^2 = 2x^2}}}<br>
The maximum area of the rectangle is then<br>
{{{(2x)*(sqrt(r^2-x^2)) = (2x)*(sqrt(2x^2-x^2)) = 2x*sqrt(x^2) = 2x*x = 2x^2 = r^2}}}