Question

a cone with a radius 9 cm has the same surface area as a cylinder with radius 6 cm and height 18cm. what is the height of the cone to the nearest thenth?

Answer 1

Question

Surface area of the cylinder: $S_{\text{cylinder}} = 2\pi r^2 + 2\pi rh = 2\pi \cdot 36 + 2\pi \cdot 6 \cdot 18 = 288\pi$.

Surface area of the cone: $S_{\text{cone}} = \pi r \sqrt{r^2 + h^2} = 9\pi \sqrt{81 + h^2}$.

So, we will have:

$$\begin{array}{l} 9\pi \sqrt{81 + h^2} = 288\pi \Rightarrow \\ \Rightarrow \sqrt{81 + h^2} = \dfrac{288\pi}{9\pi} = 32 \Rightarrow h^2 = 32^2 - 81 = 943 \Rightarrow h = 30.7 \text{ cm}. \end{array}$$

Answer: 30.7 cm.