Find the radius, volume, and hieght of the right- circlar cylindar that can be inscribed in a right- circular cone with a radius of 6 inches and a hieght of 10 inches.

Question

There are an infinite number of solutions to this. Did you mean the right cylinder with the max volume? Or surface area?

If it is the right cylinder of max volume. Consider this.

The height and radius of the cylinder are related:
h= 10 - 10r/6

Volume cylinder = PI * r^2 * h
= PI*r^3 *(10 - 10r/6 )

Take the derivatative, set to zero . and solve.