To find the height of a cone given its diameter and volume, you can follow these steps:
1. Understand the given information:
- Diameter of the cone (d) = 20 cm
- Volume of the cone (V) = 2094.4 cubic cm
2. Calculate the radius of the cone:
- The radius (r) is half of the diameter.
\(\[
r = \frac{d}{2} = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm}
\]\)
3. Recall the formula for the volume of a cone:
The volume \(\( V \)\) of a cone is given by:
\(\[
V = \frac{1}{3} \pi r^2 h
\]\)
Where:
- \(\( V \)\) is the volume
- \(\( r \)\) is the radius
- \(\( h \)\) is the height
- \(\( \pi \)\) is a constant (approximately 3.14159)
4. Rearrange the volume formula to solve for height (h):
\(\[
V = \frac{1}{3} \pi r^2 h
\]\)
Solving for height \(\( h \)\):
\(\[
h = \frac{3V}{\pi r^2}
\]\)
5. Substitute the known values into the formula:
\(\[
h = \frac{3 \times 2094.4 \, \text{cm}^3}{\pi \times (10 \, \text{cm})^2}
\]\)
6. Simplify the equation:
\(\[
h = \frac{6283.2 \, \text{cm}^3}{\pi \times 100 \, \text{cm}^2}
\]\)
\(\[
h = \frac{6283.2}{314.159} \, \text{cm} \quad \text{(approximately)}
\]\)
7. Calculate the height:
- Using the approximation \(\(\pi \approx 3.14159\)\), we divide:
\(\[
h \approx 20.00004676869994 \, \text{cm}
\]\)
8. Conclusion:
- The height of the cone is approximately \(\(20.00004676869994\)\) cm.
Therefore, the height of the cone given the diameter of 20 cm and volume of 2094.4 cubic cm is approximately \(\(20.00004676869994\)\) cm.
