Question

A cylinder whose height is 3 times its radius is inscribed in a cone whose height

is 6 times its radius. What fraction of the cone’s volume lies inside the cylinder?
Express your answer as a common fraction.

Answer 1

The fraction of the cone's volume that lies inside the cylinder is 1/2.

Step-by-step explanation:

To find the fraction of the cone's volume that lies inside the cylinder, we need to compare their volumes. The formula for the volume of a cylinder is V = π r² h, where r is the radius and h is the height. Given that the height of the cylinder is 3 times its radius, we can calculate its volume as V = π (r)² (3r) = 3π r³. Similarly, the volume of the cone is given by V = π r² (6r) = 6π r³. Therefore, the fraction of the cone's volume that lies inside the cylinder is 3π r³ / 6π r³ = 0.5, which simplifies to 1/2.

Answer 2

Step-by-step explanation:

Let the radius of the cylinder be $r$ and the radius of the base of the cone be $R$. Then the height of the cylinder is $3r$ and the height of the cone is $6R$. Since the cylinder is inscribed in the cone, the radius of the cylinder is equal to the radius of the base of the cone.

The volume of the cylinder is $V_{cyl}=\pi r^2(3r)=3\pi r^3$, and the volume of the cone is $V_{cone}=\frac13\pi R^2(6R)=2\pi R^3$. We want to find the ratio of the volume of the cylinder to the volume of the cone.

Since the radius of the cylinder is equal to the radius of the base of the cone, we can relate $R$ and $r$ using similar triangles: $$\frac{R}{3R}=\frac{r}{3r}\quad\Rightarrow\quad r=\frac{R}{3}.$$ Substituting this into the expressions for the volumes, we get $$\frac{\text{Volume of cylinder}}{\text{Volume of cone}}=\frac{3\pi(\frac{R}{3})^3}{2\pi R^3}=\frac{1}{6}.$$ Therefore, $\frac{1}{6}$ of the volume of the cone lies inside the cylinder.