Question
How do I find the correct height of a right circular cone with a given height of 15 cm, diameter of 13 cm and slant height of 10 cm?
Answers
GPT-4o mini
To find the correct height of a right circular cone, we can use the relationship between the radius \( r \), height \( h \), and slant height \( l \). The formula for a right circular cone is based on the Pythagorean theorem:
\[
l^2 = r^2 + h^2
\]
Where:
- \( l \) is the slant height
- \( r \) is the radius
- \( h \) is the height
Given the values:
- Diameter = 13 cm, so the radius \( r = \frac{13}{2} = 6.5 \) cm
- Height \( h = 15 \) cm (you mentioned a height, but we will check if this is consistent with the slant height)
- Slant height \( l = 10 \) cm
Now, let's plug \( r \) and \( l \) into the equation to see if the height of 15 cm is consistent:
1. Calculate \( l^2 \):
\[
l^2 = 10^2 = 100
\]
2. Calculate \( r^2 \):
\[
r^2 = (6.5)^2 = 42.25
\]
3. Now substitute \( l^2 \) and \( r^2 \) into the Pythagorean theorem:
\[
l^2 = r^2 + h^2 \implies 100 = 42.25 + h^2
\]
4. Solve for \( h^2 \):
\[
h^2 = 100 - 42.25 = 57.75
\]
5. Now take the square root to find \( h \):
\[
h = \sqrt{57.75} \approx 7.6 \, \text{cm}
\]
So the calculated height of the cone is approximately \( 7.6 \) cm, indicating that the given height of 15 cm does not match up with the other dimensions provided (diameter and slant height).
Therefore, if you're looking for the correct height based on the given diameter and slant height, the height should be approximately 7.6 cm, not 15 cm.
\[
l^2 = r^2 + h^2
\]
Where:
- \( l \) is the slant height
- \( r \) is the radius
- \( h \) is the height
Given the values:
- Diameter = 13 cm, so the radius \( r = \frac{13}{2} = 6.5 \) cm
- Height \( h = 15 \) cm (you mentioned a height, but we will check if this is consistent with the slant height)
- Slant height \( l = 10 \) cm
Now, let's plug \( r \) and \( l \) into the equation to see if the height of 15 cm is consistent:
1. Calculate \( l^2 \):
\[
l^2 = 10^2 = 100
\]
2. Calculate \( r^2 \):
\[
r^2 = (6.5)^2 = 42.25
\]
3. Now substitute \( l^2 \) and \( r^2 \) into the Pythagorean theorem:
\[
l^2 = r^2 + h^2 \implies 100 = 42.25 + h^2
\]
4. Solve for \( h^2 \):
\[
h^2 = 100 - 42.25 = 57.75
\]
5. Now take the square root to find \( h \):
\[
h = \sqrt{57.75} \approx 7.6 \, \text{cm}
\]
So the calculated height of the cone is approximately \( 7.6 \) cm, indicating that the given height of 15 cm does not match up with the other dimensions provided (diameter and slant height).
Therefore, if you're looking for the correct height based on the given diameter and slant height, the height should be approximately 7.6 cm, not 15 cm.