Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 105 in.Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail.What is the volume of such a package? Hint: The length plus the girth is , and the volume is .

Question

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 105 in.Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail.What is the volume of such a package? Hint: The length plus the girth is  , and the volume is  .

Quizplus · Accepted Answer

Let the radius of the cylinder be $r$ and let the height of the cylinder be $h$. Then, we have the following constraints:

$2\pi r + 2h \leq 105$ (since the maximum combined length and girth is 105)
$V = \pi r^2h$
We want to maximize $V$. We can first use the first constraint to solve for $h$ in terms of $r$:
\begin{align*}
2\pi r + 2h &= 105 \
h &= \frac{105 - 2\pi r}{2}
\end{align*}
Substituting into the formula for $V$:
\begin{align*}
V &= \pi r^2h \
&= \pi r^2\left(\frac{105 - 2\pi r}{2}ight) \
&= \frac{\pi}{2}(105r^2 - 2\pi r^3) \
&= \frac{\pi}{2}(2r^2)(\frac{105}{2} - \pi r)
\end{align*}
So, in order to maximize $V$, we need to maximize $(\frac{105}{2} - \pi r)$.
\begin{align*}
\frac{d}{dr}\left(\frac{105}{2}-\pi right) &= -\pi \
\frac{d}{dr}(r^2) &= 2r
\end{align*}
Setting these derivatives equal to zero, we find that the maximum occurs when $r = \frac{105}{2\pi}$ and $h = \frac{105 - 2\pi r}{2}$.
The volume of the cylinder is then:
\begin{align*}
V &= \frac{\pi}{2}\left(\frac{105}{2\pi}ight)^2\left(\frac{105 - 2\pi (\frac{105}{2\pi})}{2}ight) \
&= \frac{\pi}{2}\left(\frac{105}{2\pi}ight)^2\left(\frac{105}{2} - \frac{105}{2}ight) \
&= \frac{2205}{4\pi} \approx 175.57
\end{align*}
Therefore, the dimensions of the cylindrical package of greatest volume that may be sent through the mail is $\frac{105}{2\pi}$ by $105-2\pi(\frac{105}{2\pi})$, and the volume of such a package is $\frac{2205}{4\pi}$. The closest answer choice is B.

Postal Regulations Specify That a Parcel Sent by Parcel Post