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 141 inches. Find the dimensions of the package of maximum volume that can be sent. (The cross section is circular.)

radius = ? in.
length = ? in.

Answer 1

radius r = 47/π in

length l = 23.5 in

Explanation:

Let us recall that:

Volume of a cylinder =
`\pi r^2 h` --- (1)

The girth g =
`2 \pi r + h` ----- (2)

Given that:

height (h) + girth (g) = 141

Then g = 141 - h ----- (3)

Equating equation (2) and (3), we have:

141 - h = 2πr + h

2h = 141 - 2πr

h = 70.5 - πr ------ (4)

From, here now, we can now replace the value of h into equation (1)

i.e.

V = πr²(70.5 - πr)

V = 70.5πr² - π²r³

Taking the differential of the above equation with respect to r, we have:

`(dv)/(dr) = 141 \pi r- 3\pi ^2 r^2`

By further differentiation:

`(d^2v)/(dr^2) = 141 \pi - 6\pi ^2 r`

Let set
`(dv)/(dr)= 0`, Then:

141πr - 3π²r² = 0

141πr = 3π²r²

Divide both sides by πr

141 = 3πr

r = 141/3π

r = 47/π in

Replacing the value of r = 47/π into equation (4), we have:

h = 70.5 - πr

h = 70.5 - π(47/π)

h = 70.5 - 47

h = 23.5 in

From equation (3);

h + g = 141

23.5 + g = 141

g = 141 - 23.5

g = 117.5 in

Volume = πr²h

V = π × (47/π )² × 23.5

V = 16523.94 in³