We are tasked with finding the height ($h$) of the cone using the formula for the volume of a cone:

$\text{Volume of a cone} = \frac{1}{3} \pi r^2 h$

Step 1: Extract the given information

• Volume ($V$) = $477\pi \, \text{cm}^3$
• Radius ($r$) = $22 \, \text{cm}$

Step 2: Substituting into the formula

$477\pi = \frac{1}{3} \pi (22)^2 h$

Simplify $(22)^2$: $477\pi = \frac{1}{3} \pi (484) h$

Step 3: Cancel $\pi$ on both sides

$477 = \frac{1}{3} \cdot 484 \cdot h$

Step 4: Simplify further

$477 = \frac{484h}{3}$

Multiply both sides by 3: $1431 = 484h$

Step 5: Solve for $h$

$h = \frac{1431}{484} \approx 2.96 \, \text{cm}$

Final Answer:

The height of the door stopper is 2.96 cm, or 3.0 cm to 1 decimal place (1 d.p.).

Would you like a detailed step-by-step breakdown of this or help with related questions?

Here are 5 follow-up questions to extend your understanding:

1. How would the volume change if the radius were doubled?
2. Can you derive the formula for the surface area of a cone with this radius and height?
3. How does the height change if the volume were increased to $1000\pi \, \text{cm}^3$?
4. What is the slant height of the cone using this radius and height?
5. Can you find the ratio of height to radius in this cone?

Tip: Always double-check units when solving geometry problems to ensure they align!