Asked by Jess

A parcel delivery service a package only of the length plus girth (distance around) does not exceed 24 inches.

A) Find the dimensions of a rectangular box with square ends that satisfies the delivery service's restriction and has a maximum volume. What is the maximum volume?

B) Find the dimensions (radius and height) of a cylindrical container that meets the delivery service's restriction and has a maximum volume. What is the maximum volume?

Answers

Answered by MathMate
Let the size of the square (cross-section) be s.
Then we need to maximize
V=s²(24-2s) with respect to s.
First find the derivative and equate to zero:
dV/ds = 48s-6s²=0
means
s=0 or s=8
s=0 corresponds to a minimum volume and
s=8 corresponds to a maximum volume.
So the maximum volume is given by
8x8x8 inches, as is evident by intuition.
Answered by Steve
but an 8x8x8 box has length+girth = 8+32 = 40 inches, so it will not work.

We need to optimize s^2(24-4s) since a square has 4 sides.

v = 24s^2 - 4s^3
v' = 48s - 12s^2
v'=0 when s=4

So, a 4x4x8 box has max volume.

Do (B) similarly
Answered by MathMate
Good catch, Steve.
Thank you.
There are no AI answers yet. The ability to request AI answers is coming soon!

Related Questions