Hector is finding the volume of a cone and the volume of a cylinder. Both the cone and the cylinder have a radius of 3 inches and a height of 5 inches. What is the volume, in cubic... Hector is finding the volume of a cone and the volume of a cylinder. Both the cone and the cylinder have a radius of 3 inches and a height of 5 inches. What is the volume, in cubic inches, of the cone? What is the ratio of the cone's volume to the cylinder's volume?
Understand the Problem
The question presents a scenario where Hector is calculating the volume of a cone and a cylinder with the same radius and height. The question asks us to calculate the volume of the cone given the radius and height, then to calculate the ratio of the cone's volume to the cylinder's volume.
Answer
Volume of the cone: $15\pi$ Ratio of the cone's volume to the cylinder's volume: $\frac{1}{3}$
Answer for screen readers
Volume of the cone: $15\pi$ cubic inches. Ratio of the cone's volume to the cylinder's volume: $\frac{1}{3}$.
Steps to Solve
- Find the volume of the cone
The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$, where $r$ is the radius and $h$ is the height. We are given $r = 3$ inches and $h = 5$ inches. Substitute these values into the formula:
$$V = \frac{1}{3}\pi (3^2)(5)$$
$$V = \frac{1}{3}\pi (9)(5)$$
$$V = \frac{1}{3}\pi (45)$$
$$V = 15\pi$$
The volume of the cone is $15\pi$ cubic inches.
- Find the volume of the cylinder
The formula for the volume of a cylinder is $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height. We are given $r = 3$ inches and $h = 5$ inches. Substitute these values into the formula:
$$V = \pi (3^2)(5)$$
$$V = \pi (9)(5)$$
$$V = 45\pi$$
The volume of the cylinder is $45\pi$ cubic inches.
- Find the ratio of the cone's volume to the cylinder's volume
The volume of the cone is $15\pi$ and the volume of the cylinder is $45\pi$. The ratio of cone's volume to cylinder's volume is:
$$\frac{\text{Volume of cone}}{\text{Volume of cylinder}} = \frac{15\pi}{45\pi} = \frac{15}{45} = \frac{1}{3}$$
The ratio of the cone's volume to the cylinder's volume is $\frac{1}{3}$.
Volume of the cone: $15\pi$ cubic inches. Ratio of the cone's volume to the cylinder's volume: $\frac{1}{3}$.
More Information
The volume of a cone is always $\frac{1}{3}$ the volume of a cylinder, given that they both share the same height and radius.
Tips
A common mistake is forgetting to multiply by $\frac{1}{3}$ when calculating the volume of the cone.
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