Determine the quadratic function whose graph is given.
-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 60 feet of fencing
To go around the garden, what dimensions will give him the maximum area in the garden?
A) width $= \frac { 120 } { \pi + 4 } \approx 16.8$, length $= 8.4$
B) width $= \frac { 60 } { \pi + 4 } \approx 8.4$, length $= 16.8$
C) width $= \frac { 120 } { \pi + 4 } \approx 16.8$, length $= 21.6$
D) width $= \frac { 120 } { \pi + 8 } \approx 10.8$, length $= 16.1$

Question

Determine the quadratic function whose graph is given.
-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 60 feet of fencing
To go around the garden, what dimensions will give him the maximum area in the garden? 
A) width $= \frac { 120 } { \pi + 4 } \approx 16.8$, length $= 8.4$
B) width $= \frac { 60 } { \pi + 4 } \approx 8.4$, length $= 16.8$
C) width $= \frac { 120 } { \pi + 4 } \approx 16.8$, length $= 21.6$
D) width $= \frac { 120 } { \pi + 8 } \approx 10.8$, length $= 16.1$

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 equation for the perimeter is $P = 2l + w + \frac{\pi w}{2} = 60$. Solving for $l$ in terms of $w$ gives $l = \frac{60 - w - \frac{\pi w}{2}}{2}$. To maximize the area $A = lw + \frac{\pi w^2}{4}$, we differentiate with respect to $w$ and find the critical points. The width that maximizes the area is $w = \frac{120}{\pi + 4}$, which makes the length $l = \frac{w}{2}$ due to the way $l$ is defined in terms of $w$.

Determine the Quadratic Function Whose Graph Is Given =120π+4≈16.8= \frac { 120 } { \pi + 4 } \approx 16.8=π+4120​≈16.8

Determine the Quadratic Function Whose Graph Is Given $= \frac { 120 } { \pi + 4 } \approx 16.8$