To find the slant height of the cone, we'll use the formula for the lateral area of a cone and solve for the slant height.
Given:
- Lateral area of the cone [tex]\( A = 60 \text{ cm}^2 \)[/tex]
- Radius of the base of the cone [tex]\( r = 6 \text{ cm} \)[/tex]
The formula for the lateral area of a cone is:
[tex]\[ A = \pi r l \][/tex]
where
[tex]\( A \)[/tex] = lateral area,
[tex]\( r \)[/tex] = radius, and
[tex]\( l \)[/tex] = slant height.
We need to solve for [tex]\( l \)[/tex]. Rearranging the formula to solve for [tex]\( l \)[/tex]:
[tex]\[ l = \frac{A}{\pi r} \][/tex]
Substitute the known values into the formula:
[tex]\[ l = \frac{60}{\pi \times 6} \][/tex]
Simplify the expression:
[tex]\[ l = \frac{60}{6\pi} \][/tex]
[tex]\[ l = \frac{10}{\pi} \][/tex]
Calculating this further, we get:
[tex]\[ l \approx 3.183098861837907 \][/tex]
Hence, the slant height [tex]\( l \)[/tex] is approximately [tex]\( 3.18 \text{ cm} \)[/tex]. Given this, none of the provided options exactly match the calculated value. However, if we were rounding differently or there was an option that should have been included, we should accurately note the value is nearest:
- 3.2 cm in approximate rounded terms.
So, the slant height is best approximated as [tex]\( 3.18 \text{ cm} \)[/tex].
