Question

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area when the total perimeter is 16 feet.

Answer 1

Answer:

Step-by-step explanation:

Given Perimeter of window

$P=16\ ft$

Suppose x and y be the dimension of rectangular window

Perimeter of rectangular portion $=x\cdot y$

Perimeter of semicircle$=\pi (0.5x)$

so

$2(x+y)+0.5\pi x=16$

$y=8-0.25\pi x-x$

Area of the window is

$A=xy+\frac{\pi x^2}{4}$

$A=x(8-0.25\pi x-x)+0.25\pi x^2$

differentiate A w.r.t x to get maximum or minimum value

$\frac{\mathrm{d} A}{\mathrm{d} x}=8-(0.5\pi +2)x+\frac{\pi x}{2}$

put $\frac{\mathrm{d} A}{\mathrm{d} x}=0$ to get maxima/minima

thus $x=4 ft$

for $x=4$ , $y=8-0.25\pi \cdot 4-4$

$y=0.858 ft$