The volume of frustum of a cone is 270.15 mm$^3$. The area of the bottom circle of the cone is 20 mm$^2$ and the area of the top circle of the cone is 5 mm$^2$. Find its height.

Using the frustum volume formula of one third times the height times the sum of the top area, bottom area, and the square root of their product, we substitute the given values. This gives 270.15 equals one third of the height times the quantity (20 plus 5 plus the square root of 20 times 5). The square root of 100 is 10, making the sum inside the parentheses equal to 35. Solving for the height gives 270.15 equals 35 over 3 times the height, which results in a height of 23.15 millimeters.

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