The volume of a cylinder is given by the formula V = π (r^2) h. Find the maximum value of V if r + h = 12

V = 256π unit^3.

Step-by-step explanation:

V = π (r^2) h

r + h = 12

So h = 12 - r.

Substituting for h in the formula for V:

V = π r^2 (12 - r)

V = 12πr^2 - πr^3

To find the value of r when V is a maximum we find the derivative with respect to r:

V' = 24πr - 3πr^2

This equals 0 for maximum/minimum:

24πr - 3πr^2 = 0

3πr (8 - r) = 0

r = 0 or 8. (We ignore the 0).

The second derivative is 24π - 6πr which is negative when r = 8 so r = 8 gives a maximum value for V.

Therefore the maximum value of V is π (8) ^2h

h = 12 - 8 = 4 so

Maximum V = 64*4π

V = 256π unit^3.