To find the slant height of the cone, you can use the Pythagorean theorem. In a right triangle formed by the height, the radius, and the slant height, the relationship can be described as follows:
\[ l^2 = h^2 + r^2 \]
Where:
- \( l \) is the slant height,
- \( h \) is the height,
- \( r \) is the radius.
Given that:
- Height \( h = 6 , \text{cm} \)
- Diameter \( d = 13 , \text{cm} \), so the radius \( r = \frac{d}{2} = \frac{13}{2} = 6.5 , \text{cm} \)
Now plug these values into the Pythagorean theorem:
\[ l^2 = 6^2 + (6.5)^2 \]
Calculating the squares:
\[ l^2 = 36 + 42.25 \] \[ l^2 = 78.25 \]
Now take the square root:
\[ l = \sqrt{78.25} \approx 8.8 , \text{cm} \]
So, rounding to the nearest tenth, the slant height is \( 8.8 , \text{cm} \).
The answer is 8.8 cm.
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