Question
Find the length of the cone d=10 cm hypotenuse 13 cm
Answers
GPT-4o mini
To find the length (height) of a cone given the diameter and hypotenuse, we can use the Pythagorean theorem.
1. **Determine the radius:** The diameter \(d\) is given as 10 cm, so the radius \(r\) is:
\[
r = \frac{d}{2} = \frac{10}{2} = 5 \text{ cm}
\]
2. **Use the Pythagorean theorem:** The cone forms a right triangle with the height \(h\), the radius \(r\), and the hypotenuse (slant height) \(l\) (which is 13 cm in this case). According to the Pythagorean theorem:
\[
l^2 = r^2 + h^2
\]
Plugging in the known values:
\[
13^2 = 5^2 + h^2
\]
This simplifies to:
\[
169 = 25 + h^2
\]
3. **Solve for \(h^2\):**
\[
h^2 = 169 - 25
\]
\[
h^2 = 144
\]
4. **Find \(h\):**
\[
h = \sqrt{144} = 12 \text{ cm}
\]
Thus, the height of the cone is \(12\) cm.
1. **Determine the radius:** The diameter \(d\) is given as 10 cm, so the radius \(r\) is:
\[
r = \frac{d}{2} = \frac{10}{2} = 5 \text{ cm}
\]
2. **Use the Pythagorean theorem:** The cone forms a right triangle with the height \(h\), the radius \(r\), and the hypotenuse (slant height) \(l\) (which is 13 cm in this case). According to the Pythagorean theorem:
\[
l^2 = r^2 + h^2
\]
Plugging in the known values:
\[
13^2 = 5^2 + h^2
\]
This simplifies to:
\[
169 = 25 + h^2
\]
3. **Solve for \(h^2\):**
\[
h^2 = 169 - 25
\]
\[
h^2 = 144
\]
4. **Find \(h\):**
\[
h = \sqrt{144} = 12 \text{ cm}
\]
Thus, the height of the cone is \(12\) cm.