We are tasked with finding the height of the cone-shaped door stopper, given its volume and radius.

The formula for the volume of a cone is:

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

Given:

• $V = 267\pi \, \text{cm}^3$
• $r = 14 \, \text{cm}$

Step 1: Plug in the known values

$267\pi = \frac{1}{3} \pi (14)^2 h$

Step 2: Simplify the equation

$267\pi = \frac{1}{3} \pi (196) h$

$267\pi = \frac{196\pi}{3} h$

Step 3: Cancel out $\pi$ from both sides

$267 = \frac{196}{3} h$

Step 4: Solve for $h$

Multiply through by 3 to eliminate the fraction: $3 \times 267 = 196h$

$801 = 196h$

Divide both sides by 196: $h = \frac{801}{196}$

$h \approx 4.1 \, \text{cm} \, (\text{to 1 decimal place})$

Final Answer:

The height of the door stopper is approximately 4.1 cm.

Would you like a step-by-step explanation or details about related concepts?
Here are 5 additional questions that build on this problem:

1. What is the relationship between the volume of a cone and its radius?
2. How would the height change if the radius were doubled while keeping the volume constant?
3. What are the units of the volume of a cone, and how are they derived?
4. How does the formula for the volume of a cone compare to that of a cylinder?
5. What are some practical applications of the cone volume formula?

Tip: Always check your units when solving geometry problems to ensure consistency!