If the volume of a cone is 1000 π/3 and the diameter is 20 mm, then the height is?
10 mm
Cone Height Calculation from Volume and Diameter
This problem asks us to find the height of a cone given its volume and diameter. We can solve this by using the formula for the volume of a cone and substituting the given values.
Given Information:
- Volume of the cone, $V = \frac{1000\pi}{3}$
- Diameter of the cone = 20 mm
Calculations:
First, we need to find the radius ($r$) of the cone. The radius is half of the diameter:
Radius, $r = \frac{\text{Diameter}}{2} = \frac{20 \text{ mm}}{2} = 10 \text{ mm}$
The formula for the volume ($V$) of a cone is:
$$V = \frac{1}{3}\pi r^2 h$$Where $r$ is the radius and $h$ is the height.
Now, we substitute the given volume and the calculated radius into the formula:
$$ \frac{1000\pi}{3} = \frac{1}{3}\pi (10 \text{ mm})^2 h $$
Let's simplify the equation:
$$ \frac{1000\pi}{3} = \frac{1}{3}\pi (100 \text{ mm}^2) h $$
To solve for the height ($h$), we can first multiply both sides of the equation by 3:
$$ 1000\pi = \pi (100 \text{ mm}^2) h $$
Next, we can divide both sides by $\pi$:
$$ 1000 = (100 \text{ mm}^2) h $$
Finally, divide both sides by $100 \text{ mm}^2$ to find the height:
$$ h = \frac{1000}{100 \text{ mm}^2} $$
$$ h = 10 \text{ mm} $$
Conclusion:
The height of the cone is 10 mm.
Answer Options Analysis:
The calculated height is 10 mm, which corresponds to the first option provided.
- 1. 10 mm - Correct
- 2. 20 mm
- 3. 200 π mm
- 4. 400 π mm
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