Given that the volume of a cylinder is 250 cm^3, and the surface area must be less than or equal to 400 cm^2, how do you find the radius and height in order for the surface area to be minimal without using derivatives?

Volume, V= πr²h

Given volume = 250 π

250 π= πr²h

r²h=250

h= 250/r²

Surface area, A = 2πr²+2πrh

A= 2πr²+ 2πr(250/r²)

A= 2πr²+ (500π/r)

dA/dr = 4πr+500π(-1/r²)

For A to be maximum or minimum, dA/dr = 0

4πr-500(π/r²) = 0

r = 5

h = 250/(5)² = 10

By second derivative test,

d²A/dr² = 4π-500π(-2/r³)

At r= 5, d²A/dr² = 4π+1000(π/r³) > 0 . It is a local minimum.

Ans: Radius = 5 and Height = 10