To calculate the height of the cone-shaped door stopper, we will use the formula for the volume of a cone:

$V = \frac{1}{3} \pi r^2 h$

Given:

• The volume $V = 477\pi \, \text{cm}^3$
• The radius $r = 22 \, \text{cm}$

Step 1: Set up the formula with the known values

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

Step 2: Simplify the equation

Cancel out $\pi$ from both sides: $477 = \frac{1}{3} \times (22)^2 \times h$ $477 = \frac{1}{3} \times 484 \times h$ $477 = \frac{484h}{3}$

Step 3: Solve for $h$

Multiply both sides by 3 to eliminate the fraction: $1431 = 484h$ Now, divide both sides by 484: $h = \frac{1431}{484} \approx 2.9566$

Step 4: Round to 1 decimal place

$h \approx 3.0 \, \text{cm}$

Thus, the height of the door stopper is 3.0 cm.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. What would be the volume of the cone if the height was doubled?
2. How would you calculate the surface area of this cone?
3. What is the ratio of the height to the radius in this cone?
4. How does changing the radius affect the volume of the cone?
5. Can you find the slant height of this cone using the Pythagorean theorem?

Tip: When working with cone volume, always remember that the height is perpendicular to the base, not along the slanted edge!