Question: curved surface area of a cone = $\pi rl$, where $r$ is the radius and $l$ is the slant height.
surface area of a sphere = $4\pi r^2$, where $r$ is the radius.
The shape below is made from a cone joined to a hemisphere.
The surface area of this shape is $156\pi$ cm²
Work out the slant height of the cone.
Graph/shape: bottom-right, a cone joined to a hemisphere with radius $6$ cm shown by a horizontal arrow across the circular join.
1. **State the problem:**
We have a shape made by joining a cone to a hemisphere. The total surface area of this shape is $156\pi$ cm². The radius of the hemisphere (and the base of the cone) is $6$ cm. We need to find the slant height $l$ of the cone.
2. **Recall formulas:**
- Curved surface area of a cone: $$A_{cone} = \pi r l$$
- Surface area of a sphere: $$A_{sphere} = 4\pi r^2$$
- Surface area of a hemisphere (half a sphere): $$A_{hemisphere} = 2\pi r^2$$
3. **Important note:**
The base of the cone is joined to the hemisphere, so the base circle of the cone is not exposed. Therefore, the total surface area of the shape is the curved surface area of the cone plus the curved surface area of the hemisphere.
4. **Set up the equation:**
Given total surface area:
$$A_{total} = A_{cone} + A_{hemisphere} = 156\pi$$
Substitute the formulas:
$$\pi r l + 2\pi r^2 = 156\pi$$
5. **Substitute $r=6$ cm:**
$$\pi \times 6 \times l + 2\pi \times 6^2 = 156\pi$$
Simplify:
$$6\pi l + 2\pi \times 36 = 156\pi$$
$$6\pi l + 72\pi = 156\pi$$
6. **Divide both sides by $\pi$ to simplify:**
$$\cancel{\pi} (6 l + 72) = \cancel{\pi} 156$$
$$6 l + 72 = 156$$
7. **Solve for $l$:**
$$6 l = 156 - 72$$
$$6 l = 84$$
$$l = \frac{84}{6}$$
$$l = 14$$
**Final answer:** The slant height of the cone is $14$ cm.