Question

1. what is the volume and surface area of this cone? write your answer in simplest fraction π form.

v=
sa=

Explanation:

Step1: Calculate the volume of the cone

The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\).
Given \(r = 4\) in and \(h=13\) in.
Substitute the values into the formula: \(V=\frac{1}{3}\pi\times4^{2}\times13=\frac{1}{3}\pi\times16\times13=\frac{208}{3}\pi\) in³.

Step2: Calculate the slant height \(l\) of the cone

Use the Pythagorean theorem \(l=\sqrt{r^{2}+h^{2}}\).
\(l=\sqrt{4^{2}+13^{2}}=\sqrt{16 + 169}=\sqrt{185}\) in.

Step3: Calculate the surface area of the cone

The formula for the surface area of a cone is \(SA=\pi r(r + l)\).
Substitute \(r = 4\) in and \(l=\sqrt{185}\) in.
\(SA=\pi\times4\times(4+\sqrt{185})=(16 + 4\sqrt{185})\pi\) in².

Answer:

\(V=\frac{208}{3}\pi\) in³, \(SA=(16 + 4\sqrt{185})\pi\) in².