Final answer:
To determine the dimensions of a Norman window that admits as much light as possible with 12 ft of framing material, we use calculus optimization. The problem involves finding a maximum, which is done by setting up an equation for the perimeter, expressing the area as a function of width, taking its derivative, and identifying critical points for maximum value.
Step-by-step explanation:
The student is seeking to maximize the area of light admitted by a Norman window, which involves optimizing the dimensions of the window given a fixed perimeter of material. This problem can be approached through the method of optimization in calculus, specifically relating to the use of derivatives to find maximum values.
For a rectangular part of the window, let the width be x feet and the length be y feet. The semicircular part has a radius of x/2 feet since its diameter would be equal to the width of the rectangle. The total perimeter of the window is then given by the perimeter of the rectangle (2x + 2y) plus the circumference of the semicircle (πx/2), which should equal 12 feet.
To optimize the area, we set up an equation for the perimeter: 2x + 2y + (πx/2) = 12, and solve for y in terms of x. We then express the area of the window as a function of x — which is x times y plus the area of the semicircle (π(x/2)2/2) — and find its derivative with respect to x. We look for critical points where the derivative is zero and use the second derivative test to identify the maximum area.