Question

For a cylinder with a surface area of 90, what is the maximum volume that it can have? Round your answer to the nearest 4 decimal places. The volume of a cylinder is πr^2*h and the surface area is 2πrh+2πr^2.

Answer 1

v = pi r^2 h

constraint is
90 = 2 pi r h + 2 pi r^2

45/pi = r h + r^2
or
h = (45/pi - r^2 )/r

so
v = pi r^2 (45/pi - r^2)/r
or
v = 45 r - pi r^3
for max or min dv/dr = 0
dv/dr = 0 = 45 - 3 pi r^2

r^2 = 15/pi
r = 2.185
now get v