What is the maximum volume of a cylinder that can be encased into a square ended cuboid whose length is 60 mm with a volume of 96 cubic cm showing work and answer to the nearest hundredth?

Answer 1

Volume of cuboid = 96 cm^3 = 96000 mm^3Length of cuboid = 60mm

Therefore cross-sectional area = 96000/60 = 1600 mm^2

Then either the cuboid is 40 mm x 40 mm with a length of 60 mm or

it is 60 mm x 60 mm square with a length of 1600/60 mm.

It is possible to prove that the largest cylinder is one whose circular face [base] is parallel to the square cross-section.

In the first case the cylinder has a radius 20 mm and height 60 mm => vol = 24000*pi mm^3, or

the cylinder is 30 mm * 30 mm and height 1600/60 mm => vol = 24000*pi mm^3.

The question contains measurements in mm and in cm and so it is not clear what units the "nearest hundredths" refers to. Since I have done the calculations in mm, the answer is

24000*pi = 75398.22 mm^3.

Answer 2

Diameter of cylinder: 96/6 = 16 and its square root is 4 cm

Maximum volume of cylinder: pi*2^squared*6 = 75.40 cubic cm rounded