Bonnie measured a cylinder with a radius of 7 centimeters and a height of 3 centimeters. What is the volume, in cubic centimeters, of the cylinder? Then, Bonnie measured a cone wit... Bonnie measured a cylinder with a radius of 7 centimeters and a height of 3 centimeters. What is the volume, in cubic centimeters, of the cylinder? Then, Bonnie measured a cone with the same height and radius as the cylinder. What is the ratio of the cylinder's volume to the cone's volume?
Understand the Problem
The question has two parts. The first part asks us to calculate the volume of a cylinder given its radius and height. The second part asks for the ratio of the cylinder's volume to the volume of a cone, given that both shapes have the same radius and height.
Answer
$147\pi$, $\frac{3}{1}$
Answer for screen readers
$147\pi$ $\frac{3}{1}$
Steps to Solve
- Calculate the volume of the cylinder
The formula for the volume of a cylinder is $V_{cylinder} = \pi r^2 h$, where $r$ is the radius and $h$ is the height. We are given $r = 7$ cm and $h = 3$ cm. Substitute the given values into the formula: $V_{cylinder} = \pi (7^2)(3)$ $V_{cylinder} = \pi (49)(3)$ $V_{cylinder} = 147\pi$ cubic centimeters
- Calculate the volume of the cone
The formula for the volume of a cone is $V_{cone} = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height. Since the cone has the same radius and height as the cylinder, $r = 7$ cm and $h = 3$ cm. Substitute the given values into the formula: $V_{cone} = \frac{1}{3} \pi (7^2)(3)$ $V_{cone} = \frac{1}{3} \pi (49)(3)$ $V_{cone} = 49\pi$ cubic centimeters
- Calculate the ratio of the cylinder's volume to the cone's volume
The ratio of the cylinder's volume to the cone's volume is: $\frac{V_{cylinder}}{V_{cone}} = \frac{147\pi}{49\pi}$ $\frac{V_{cylinder}}{V_{cone}} = \frac{147}{49}$ $\frac{V_{cylinder}}{V_{cone}} = 3$ $\frac{V_{cylinder}}{V_{cone}} = \frac{3}{1}$
$147\pi$ $\frac{3}{1}$
More Information
The volume of the cylinder is $147\pi$ cubic centimeters, and the ratio of the cylinder's volume to the cone's volume is 3:1.
Tips
A common mistake is to forget the $\frac{1}{3}$ factor in the volume of a cone formula. Another common mistake is to mix up the formulas for the volume of a cylinder and cone.
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