To find the height of the cone, we can use the relationship between the radius, height, and slant height of the cone, which can be expressed using the Pythagorean theorem.
Given:
- Diameter of the cone = 10 cm
- Therefore, the radius \( r \) = \( \frac{10}{2} = 5 \) cm
- Slant height \( l \) = 13 cm
We can use the following formula derived from the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Where:
- \( l \) = slant height
- \( r \) = radius
- \( h \) = height
Substituting the known values into the equation:
\[ 13^2 = 5^2 + h^2 \]
Calculating the squares:
\[ 169 = 25 + h^2 \]
Now, subtract 25 from both sides:
\[ 169 - 25 = h^2 \]
\[ 144 = h^2 \]
Now, take the square root of both sides to find \( h \):
\[ h = \sqrt{144} = 12 \text{ cm} \]
Thus, the height of the cone is 12 cm.
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