Let V be the volume and S the total surface area of a solid right circular cylinder that is 5 ft high and has a radius r ft. Find dV/dS when r = 3.

Answer 1

The volume of a cylinder is πr²h and the total surface area is 2πr² +2πrh

You should not have to memorize these formulas.

Assuming that you know the area of a circle is πr² and the circumference is 2πr,

the volume of a cylinder, as for a box or any prism, is Ah, where A is the base. The base of the cylinder is a circle, so A = πr² and V = Ah = πr²h

For the total surface area, the area of the top and bottom of the cylinder are circles, so the area is 2A = 2πr²

For the surface area of the side (lateral surface area), it is the perimeter of the circular cross section times the height equals the circumference times the height = 2πrh

What seems to be undefined is whether the radius is changing or the height is changing. I am assuming that the radius is changing.

Then, while we could write a formula for V in terms of S, it is easier to calculate

dV/dS = dV/dr/(dS/dr)

dV/dr = d(πr²h)/dr = 2πrh

dS/dr = d(2πr² +2πrh)/dr = 4πr + 2πh

Then, dV/dS = 2πrh/(4πr + 2πh) = rh/(2r + h)

h =5 and r= 3, so

dV/dS =3(5)/(2(3)+5) = 15/(6+5) = 15/11