The internal and external radii are estimated to be 6 cm and 8 cm respectively, to the nearest whole number. The height of the cylinder is exactly 14 cm.

Question

(a) Determine the exact values for internal and external radii which will give maximum volume of the material used.

(b) Calculate the maximum possible volume of the material used. Take the value of Pi to be 22/7

Answer 1

for a cylinder, max volume will occur when the max cross-section area is achieved.
If the radii are exactly 6 and 8, then the area is π(8^2-6^2) = 28π
If the accuracy is within ±x, then we have the area πa satisfying

a = π((8+x)^2-(6-x)^2) = 28π(x+1)
So, the only constraint is the allowable values of x. Since the radii were estimated to the nearest integer, x <= 0.5
So the radius 5.5 <= r <= 8.5
and the volume is 14π(8.5^2-5.5^2) = 588π cm^3