A "Norman window" consists of a rectangle surmounted by a semicircle whose diameter…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A "Norman window" consists of a rectangle surmounted by a semicircle whose diameter coincides with the top of the rectangle. Suppose W represents the width of the rectangle (and thus the diameter of the semicircle) and H represents the height of the rectangular portion only. A custom window designer has exactly 30 meters of framing material to outline the perimeter of a single Norman window (including the curved part of the semicircle).
Complete the tasks below.
(a) Express the total area A of the window in terms of W and H.
(b) Using the perimeter constraint, rewrite A as a function of a single variable.
(c) Determine all critical numbers of the area function found in part (b). Provide exact values if possible; otherwise give decimal approximations accurate to three decimal places.
(d) Prove that the critical number identified yields the maximum possible area under the given constraint.
(e) State the optimal dimensions W and H (include units) and compute the corresponding maximum area of the window.