Answer:
V = 256π unit^3.
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.