Find the circular cylinder of greatest volume that can be inscribed in a right circular cone with altitude?

If you slice the cone you have an altitude of 20 and a diameter of 10 so drop a perpendicular to get a right triangle of height 20 and base 5. If you look at the right triangle on the right side, the hypotenuse can be viewed as a line with equation y=-4x+20 because you have the points (5,0) from the radius and (0,20) from the height. So for any x between 0 and 5, the corresponding height will be -4x+20. Maximizing the area of this rectangle also maximizes the volume of the circular cylinder. The area is A=x* (-4x+20). When you find A' and set it to zero and solve, you get x=2.5 cm.