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:

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