Solve the Problem $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + 3 \pi }$

Multiple Choice

Solve the problem.
-A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only one-fourth as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

A) $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + 3 \pi }$
B) $\frac { \text { width } } { \text { height } } = \frac { 4 } { 8 + 3 \pi }$
C) $\frac { \text { width } } { \text { height } } = \frac { 16 } { 8 + \pi }$
D) $\frac { \text { width } } { \text { height } } = \frac { 16 } { 4 + 3 \pi }$