Answer
a) $A(x)=-0.895x^2+15x$
b) $8.38 ft:4.23 ft$ (with $x=8.38$)
Work Step by Step
a) Let's say $\pi=3.14$ (for simpler calculations).
To find the function of the window's area, we must rely on information about perimeter to find necessary information about lengths of the sides.
The perimeter of the semicircle is: $\frac{1}{2}x\pi=\frac{3.14}{2}x=1.57x$ (because x is the semicircle's diameter, as showed in the image).
To find the perimeter, we need to add up the perimeter of the semicircle -- the length of the bottom side and the length of two rear sides. We already know the first two in terms of $x$, and we also know that the subsequent perimeter of the entire window is $30$ ft. Therefore, one rear side is $\frac{30-x-1.57x}{2}$ ft, or $15-1.285x$ in length.
Now, we can construct the function of the window's area.
The area of the semicircle is: $\frac{1}{2}(\frac{1}{2}x)^2\pi=\frac{1}{8}*3.14x^2=0.39x^2$
The area of the rectangular part is: $x(15-1.285x)=-1.285x^2+15x$
Thus:
$A(x)=0.39x^2-1.285x^2+15x$
$A(x)=-0.895x^2+15x$
(with $a=-0.895$, $b=15$, $c=0$)
b) For a window to admit a largest amount of light, its area must be as large as possible. Therefore, we can use our minimum/maximum formula to solve this problem.
The maximum value occurs at:
$x=\frac{-b}{2a}=\frac{-15}{2*(-0.895)}=8.38$
The question asks us to solve for the dimension of the window at maximum allowance, so we don't need to calculate its maximum area. Instead, we are meant to find the side lengths of the rectangular part. The bottom side's length is already known, as it equals to $x$ itself. To calculate the rear side's length, we use the expression we have derived in part (a)
Rear$=15-1.285x=15-1.285*8.38=4.23$ ft
