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 the height be $h$. Then the combined length and girth is $2\pi r + 2h \leq 105$, or $\pi r + h \leq 52.5$. We want to maximize the volume, which is $\pi r^2 h$. We can solve for $h$ in terms of $r$ using the inequality: $h \leq 52.5 - \pi r$, so $V(r) = \pi r^2 (52.5 - \pi r)$. We can take the derivative and set it equal to zero to find the maximum:
$$V'(r) = 2\pi r (52.5 - 2\pi r) = 0 \quad \Rightarrow \quad r = \frac{52.5}{2\pi}.$$
Plugging this back into the inequality, we find the maximum height $h = \frac{105}{2}-\pi r$. Therefore, the dimensions of the package of greatest volume are $2r$ (diameter) by $h$, or $\frac{105}{\pi}$ by $\frac{105}{2\pi}-\frac{105}{2\pi} = \frac{105}{2\pi}$ inches. The volume is then $V = \pi r^2 h = \frac{105^3}{4\pi^2}$ cubic inches or about $\boxed{1100}$ cubic inches rounded to the nearest integer.

Postal Regulations Specify That a Parcel Sent by Parcel Post