The volume of a cone is 113.04 cubic millimeters, with a radius of 3mm. Using $\pi \approx 3.14$, what is the height of this cone? Round your answer to the nearest hundredth.
Understand the Problem
We are given the volume and the radius of a cone and need to find the height. We will use the formula for the volume of a cone, $V = \frac{1}{3}\pi r^2 h$, and solve for h.
Answer
$h = 12.00$
Answer for screen readers
$h = 12.00$ millimeters
Steps to Solve
- Write the formula for the volume of a cone
The formula for the volume of a cone is: $V = \frac{1}{3}\pi r^2 h$
- Plug in the given values
We are given that the volume $V = 113.04$ cubic millimeters and the radius $r = 3$ millimeters. We also know that we are using $\pi = 3.14$. Plug these values into the formula: $113.04 = \frac{1}{3}(3.14)(3^2)h$
- Simplify the equation
Simplify the right side of the equation: $113.04 = \frac{1}{3}(3.14)(9)h$ $113.04 = (3.14)(3)h$ $113.04 = 9.42h$
- Solve for h Divide both sides by 9.42 to isolate $h$: $h = \frac{113.04}{9.42}$ $h = 12$
$h = 12.00$ millimeters
More Information
The height of the cone is exactly 12 millimeters, so rounding to the nearest hundredth does not change the value.
Tips
A common mistake is to forget the $\frac{1}{3}$ factor in the volume of a cone formula. Also, errors can occur during the arithmetic simplification steps.
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