Question

find the volume and surface area of the cone if the height is 17m and the radius is 5m

Explanation:

Step1: Calculate the slant height $l$

Use the Pythagorean theorem $l=\sqrt{h^{2}+r^{2}}$, where $h = 17$ and $r = 5$. So $l=\sqrt{17^{2}+5^{2}}=\sqrt{289 + 25}=\sqrt{314}$.

Step2: Calculate the volume $V$ of the cone

The formula for the volume of a cone is $V=\frac{1}{3}\pi r^{2}h$. Substitute $r = 5$ and $h=17$. Then $V=\frac{1}{3}\pi\times5^{2}\times17=\frac{425\pi}{3}$.

Step3: Calculate the surface - area $S$ of the cone

The formula for the surface area of a cone is $S=\pi r(r + l)$. Substitute $r = 5$ and $l=\sqrt{314}$. So $S=\pi\times5\times(5+\sqrt{314})=5\pi(5 + \sqrt{314})$.

Answer:

Volume: $\frac{425\pi}{3}$; Surface Area: $5\pi(5+\sqrt{314})$