Question

A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 144 inches. Find the dimensions of the package of maximum volume that can be sent. (The cross section is circular.)

Answer 1

Answer:

The dimensions of the package is $r=\frac{48}{\pi}\ \text{and} \ h=48$.

Step-by-step explanation:

Consider the provided information.

As it is given that, cylindrical package to be sent by a postal service can have a maximum combined length and girth is 144 inches.

Therefore,

144 = 2$\pi$r + h

144-2$\pi$r = h

The volume of a cylindrical package can be calculated as:

$V=\pi r^{2}h$

Substitute the value of h in the above equation.

$V=\pi r^{2}(144-2\pi r)$

Differentiate the above equation with respect to r.

$\frac{dV}{dr}=2\pi r(144-2\pi r)+\pi r^{2}(-2\pi)$

$\frac{dV}{dr}=288\pi r-4{\pi}^2 r^{2}-2{\pi}^2 r^{2}$

$\frac{dV}{dr}=288\pi r-6{\pi}^2 r^{2}$

$\frac{dV}{dr}=-6\pi r(-48+\pi r)$

Substitute $\frac{dV}{dr}=0$ in above equation.

$0=-6\pi r(-48+\pi r)$

Therefore,

$0=-48+\pi r$

$r=\frac{48}{\pi}$

Now, substitute the value of r in 144-2$\pi$r = h.

$144-2\pi\frac{48}{\pi}=h$

$144-96=h$

$48=h$

Therefore the dimensions of the package should be:

$r=\frac{48}{\pi}\ \text{and} \ h=48$