The volume of a cylinder is given by the formula V = π r2h. Find the maximum value of V if r + h = 12.

Question

PLEASE HELP, The topic is Applications to maximum and minimum problems

Answer 1

V = π * r² * h, and r + h = 12

You have 2 independent variables, so use both equations to develop a formula for V. It seems best to use substitution for h, to keep the equation relatively simple....So:

r + h = 12

rearrange to get: h = 12 - r

Substitute in the 1st equation to get:

V = π * r² * (12 - r)

Expand:

V = 12πr² - πr³

To find a maximum (or minimum), the typical method is to find the 1st derivative and set it equal to zero.

V' = 24πr - 3πr²

Set equal to 0 and solve for r:

24πr - 3πr² = 0

8r - r² = 0

r = 0, 8 (Interpretation: 0 gives you the minimum volume, and 8 gives you the maximum)

CHECK this by plugging into the original formula and then trying values on either side.

Answer 2

V = πr²h

Since r + h = 12, h = 12 - r

So, V = πr²(12 - r) = π(12r² - r³)

dV/dr = π(24r - 3r²) = 3πr(8 - r) = 0 when r = 0 or r = 8

When 0 < r < 8, dV/dr > 0, so V is increasing

When r > 8, dV/dr < 0, so V is decreasing

There is a relative (and absolute) max when r = 8.

Maximum volume = π(8)²(12 - 8) = 256π cubic units

Answer 3

Elizabeth,

The volume V is a function of 2 variables r and h. The problem tells you that r+h=12 so if we solve this for h we have h=12-r we can the substitute this in the equation for V. i.e. V=πr²(12-r) so now you have V asa function of one variable r. You can now differentiate V with respect to r and set equal to 0 and solve for the value of r.

Is this enough for you to get the problem solved?

Jim

Answer 4

Hello Elizabeth,

Please watch my video to see the answer.

It is helpful as well to graph V in terms of r when r+h = 12 so that you can see that V is at a maximum when r = 8.