Question 1116692
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I can't see your figure; so I guessed that x is the width of the rectangle (and therefore the diameter of the semicircle).  And I used y for the height of the rectangle.<br>
We want to find the value of x that makes the greatest possible amount of light admitted through the window, given that the perimeter of the window is 32 feet.  That means we want to maximize the area of the window.<br>
So we need an expression in terms of a single variable (preferably x) for the area of the window.<br>
The perimeter of the window, 32 feet, is the width of the rectangle, plus twice the height of the rectangle, plus the circumference of the semicircle:<br>
{{{x+2y+(pi)(x/2) = 32}}}<br>
The area of the window is {{{xy+(1/2)(pi)(x/2)^2 = xy+(pi)x^2/8}}}<br>
We can solve the equation for the perimeter for y in terms of x and substitute into the formula for the area to get the area in terms of x only.<br>
{{{2y = 32-(1+pi/2)x = 32-((2+pi)/2)x}}}
{{{y = 16-((2+pi)/4)x}}}<br>
Then the area of the window in terms of x only is<br>
{{{x(16-((2+pi)/4)x)+(pi)x^2/8}}}
{{{16x-((2+pi)/4)x^2+(pi)x^2/8}}}<br>
We differentiate and set the derivative equal to zero to find the value of x that maximizes the area.<br>
{{{16 - 2x((2+pi)/4)+(pi)x/4 = 0}}}
{{{16 - x - (pi)x/2 + (pi)x/4 = 0}}}
{{{16 = x(1+(pi)/4)}}}
{{{x = 16/(1+(pi)/4)}}} = 8.96 ft, to 2 decimal places