A Norman window has the outline of a semicircle on top of a rectangle. Suppose there is feet of wood trim available. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window.
A) base: ft; height: ft
B) base: ft; height: ft
C) base: ft; height: ft
D) base: ft; height: ft

Question

A Norman window has the outline of a semicircle on top of a rectangle. Suppose there is  feet of wood trim available. Find the dimensions of the rectangle (and, hence, the semicircle) that will maximize the area of the window.
A) base: ft; height: ft 
B) base: ft; height: ft 
C) base: ft; height: ft 
D) base: ft; height: ft

Quizplus · Accepted Answer

Let the rectangle have a base of length x and height y, and let r be the radius of the semicircle. Then the total wood trim used is x+2y+πr. We want to maximize the area of the window, which is xy+πr^2/2. Since we have a constraint on the amount of wood trim used, we can solve for one of the variables in terms of the other and substitute into the area formula. For example, solving for r gives r=(x/2)+y/2. Substituting this into the area formula results in A(x,y)=xy+(π/8)(x^2+4y^2). This is a function of two variables, so we can find critical points by taking the partial derivatives and setting them equal to zero: ∂A/∂x=y+(π/4)x=0 and ∂A/∂y=x+(π/2)y=0. Solving these simultaneous equations gives x=(2/π)y and y=(4/π^2)x. Substituting either of these into the constraint x+2y+πr=11 gives the value for the corresponding variable: x=11/(2+π) and y=11(4π-2π^2)/(π^2+4). Plugging these into the area formula gives A≈8.45. Note that this is a local maximum since the second partial derivatives of A with respect to x and y are both positive. The answer is therefore D, with the given dimensions for x and y.

A Norman Window Has the Outline of a Semicircle on Top