A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be $58 \mathrm{ft}$. What should its dimensions be in order to allow the maximum amount of light to enter through the window?
To allow the maximum amount of light to enter through the window, the radius of the semicircle should be and the height of the rectangle should be (Do not round until the final answer. Then round to two decimal places as needed.)

Step 1 ：Let the radius of the semicircle be \(r\) and the height of the rectangle be \(h\). The perimeter of the window is the sum of the circumference of the semicircle and the height of the rectangle, which is \(2r + h + \pi r = 58\).

Step 2 ：We can solve this equation for \(h\) to get \(h = 58 - 2r - \pi r\).

Step 3 ：The area of the window is the sum of the area of the rectangle and the area of the semicircle, which is \(A = 2rh + \frac{1}{2} \pi r^2\).

Step 4 ：We can substitute \(h = 58 - 2r - \pi r\) into the area equation to get \(A = 2r(58 - 2r - \pi r) + \frac{1}{2} \pi r^2\).

Step 5 ：We can simplify this to get \(A = 116r - 4r^2 - 2\pi r^2 + \frac{1}{2} \pi r^2 = 116r - 4r^2 - \frac{3}{2}\pi r^2\).

Step 6 ：To find the maximum area, we can take the derivative of \(A\) with respect to \(r\) and set it equal to 0. This gives us \(A' = 116 - 8r - 3\pi r = 0\).

Step 7 ：Solving this equation for \(r\) gives us \(r = \frac{116}{8 + 3\pi}\).

Step 8 ：Substituting this value of \(r\) back into the equation for \(h\) gives us \(h = 58 - 2(\frac{116}{8 + 3\pi}) - \pi (\frac{116}{8 + 3\pi})\).

Step 9 ：Calculating these values gives us \(r \approx 7.54\) and \(h \approx 42.92\).

Step 10 ：So, to allow the maximum amount of light to enter through the window, the radius of the semicircle should be approximately 7.54 ft and the height of the rectangle should be approximately 42.92 ft. \(\boxed{r \approx 7.54, h \approx 42.92}\)