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