To find the slant height of the cone, we can use the Pythagorean theorem. In a right triangle formed by the height, radius, and slant height of the cone, the following relationship holds:
\[ c^2 = a^2 + b^2 \]
where:
- \(c\) is the slant height,
- \(a\) is the height of the cone (6 cm),
- \(b\) is the radius of the base.
First, we need to find the radius. The diameter of the cone is given as 13 cm, so the radius \(b\) is:
\[ b = \frac{diameter}{2} = \frac{13 , \text{cm}}{2} = 6.5 , \text{cm} \]
Now, applying the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ c^2 = (6 , \text{cm})^2 + (6.5 , \text{cm})^2 \]
Calculating \(6^2\) and \(6.5^2\):
\[ c^2 = 36 + 42.25 = 78.25 \]
Now, solving for \(c\):
\[ c = \sqrt{78.25} \approx 8.83 , \text{cm} \]
Rounding to the nearest tenth, the slant height \(c\) is approximately:
\[ \boxed{8.8 , \text{cm}} \]