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# A cylinder is inscribed in a right circular cone of height 6.5 and radius (at the base) equal to 8.

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A cylinder is inscribed in a right circular cone of height 6.5 and radius (at the base) equal to 8. What are the dimensions of such a cylinder which has maximum volume?

Height =

May 3, 2022

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Suppose r is the base radius of the cylinder inscribing the cone. Let h be the height.

Using similar triangles on the largest cross-section,

$$\dfrac{h}{8 - r} = \dfrac{6.5}8\\ h = \dfrac{13}2 - \dfrac{13}{16}r$$

Then the volume is $$\pi r^2 h = \dfrac{13}{16} \pi r^2 (8 - r)$$.

Solving $$\dfrac{d}{dr} \dfrac{13}{16} \pi r^2 (8 - r) = 0$$ gives r = 16/3 or r = 0 (rej.)

Note that

$$\dfrac{d^2}{dr^2}\Big|_{r = \frac{16}3} \dfrac{13}{16} \pi r^2 (8 - r) = -13\pi < 0$$. Then maximum volume is attained when r = 16/3.

Now, you can calculate the height and get your answer.

May 3, 2022