Final answer:
To maximize the area of a window with a rectangular base and semicircular top given a fixed perimeter, one must derive the length in terms of radius, set up an area function, take its derivative, and solve for the radius before calculating the length. Comparing the results with the given options provides the correct dimensions.
Step-by-step explanation:
An architect wants to maximize the area of a window that has a rectangular shape with a semicircle on top. The perimeter of the window is given as 36 feet, and we are tasked with finding the dimensions that will maximize the window's total area. To solve this problem, we can use calculus or geometrical reasoning.
Let the width of the rectangle (and the diameter of the semicircle) be 2r, and the length of the rectangle be l. The perimeter of the rectangle is 2l + 2r, and the perimeter of the semicircle is πr. Given the total perimeter P as 36 feet, we have:
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We want to maximize the area of the window, which is the sum of the area of the rectangle lr and the area of the semicircle (πr²)/2. So our objective function to maximize is:
A = lr + (πr²)/2
Substituting l into the area equation, we get the area as a function of r. Then we take the derivative with respect to r, set it to zero, and solve for r. After finding r, we substitute back to find l. The dimensions that maximize the area will match one of the provided options.