solving calculus problems

Question

An open cylindrical tank at the top and has a volume of 100 liters will be built. If the radius of the cylinder is r metre, find the vast amount of metal needed to build it, and find the value of r so that the surface area is minimal. Find the minimum area is also used metal.

Answer 1

Surface area S=2πrh+πr² where h is the cylinder's height.

Volume V=πr²h=100L, so h=100/(πr²), and:

S=2πr(100/(πr²))+πr²=200/r+πr²=(200+πr³)/r.

Note that 1L=1000mL=1000cc=1000cm³=1000×(10^-2)³=0.001m³.

For minimum S, dS/dr=0:

-200/r²+2πr=0, -100+πr³=0,

r³=100/π=31.831L approx=31831cm³).

r=31.692cm=0.3169m approx.

This corresponds to a minimum surface area=9466cm²=0.9466m² approx.