To find the slant height of the cone, we can use the Pythagorean Theorem. The theorem states that for a right triangle:
\( a^2 + b^2 = c^2 \)
Where:
- \( a \) is the height of the cone
- \( b \) is the radius of the base of the cone
- \( c \) is the slant height of the cone
Given:
- The height (\( a \)) of the cone is 6 cm.
- The diameter of the cone is 13 cm, so the radius (\( b \)) is half of that, which is \( \frac{13}{2} = 6.5 \) cm.
Now, we can plug these values into the Pythagorean theorem.
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Square the height and the radius:
- \( a^2 = 6^2 = 36 \)
- \( b^2 = (6.5)^2 = 42.25 \)
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Now apply the Pythagorean theorem:
- \( c^2 = a^2 + b^2 \)
- \( c^2 = 36 + 42.25 = 78.25 \)
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Now solve for \( c \) (the slant height):
- \( c = \sqrt{78.25} \)
- \( c \approx 8.839 \)
Rounding to the nearest tenth, the slant height is approximately 8.8 cm.
So the answer is 8.8 cm.