Solve the problem.
-Alan is building a garden shaped like a rectangle with a semicircle attached to one short side along its diameter. The diameter of the semicircle is equal to the width of the short side of the rectangle. If he has 30 feet of fencing to go around the garden, what dimensions will give him the maximum area in the garden?
A) width $= \frac { 60 } { \pi + 4 } \approx 8.4$, length $= 4.2$
B) width $= \frac { 30 } { \pi + 4 } \approx 4.2$, length $= 8.4$
C) width $= \frac { 60 } { \pi + 4 } \approx 8.4$, length $= 10.8$
D) width $= \frac { 60 } { \pi + 8 } \approx 5.4$, length $= 8.1$

Question

Quizplus · Accepted Answer

The perimeter of the garden is the sum of the rectangle's two lengths and one width, plus the circumference of the semicircle ($\frac{\pi d}{2}$, where $d$ is the diameter or the width of the rectangle). The total fencing available is 30 feet, so $2l + w + \frac{\pi w}{2} = 30$. To maximize the area ($A = lw + \frac{\pi w^2}{8}$), we differentiate with respect to $w$ and solve, leading to the dimensions where width $w = \frac{60}{\pi + 4}$ and length $l = \frac{w}{2}$, which matches option A.

Solve the Problem =60π+4≈8.4= \frac { 60 } { \pi + 4 } \approx 8.4=π+460​≈8.4

Solve the Problem $= \frac { 60 } { \pi + 4 } \approx 8.4$