Question

1. cone

a cone has a radius of 3 cm and height 12 cm.
a. find its slant height.
b. calculate its surface area.
c. find its volume.

Explanation:

Step1: Find the slant height

The slant height \(l\) of a cone can be found using the Pythagorean theorem \(l=\sqrt{r^{2}+h^{2}}\), where \(r = 3\) cm and \(h=12\) cm.

$$
LATEXBLOCK0
$$

Step2: Calculate the surface area

The surface area \(S\) of a cone is \(S=\pi r(r + l)\). Substitute \(r = 3\) cm and \(l = 3\sqrt{17}\) cm.

$$
LATEXBLOCK1
$$

Step3: Find the volume

The volume \(V\) of a cone is \(V=\frac{1}{3}\pi r^{2}h\). Substitute \(r = 3\) cm and \(h = 12\) cm.

$$
LATEXBLOCK2
$$

Answer:

a. The slant height is \(3\sqrt{17}\text{ cm}\approx12.37\text{ cm}\).
b. The surface area is \(9\pi(1 + \sqrt{17})\text{ cm}^{2}\approx144.7\text{ cm}^{2}\).
c. The volume is \(36\pi\text{ cm}^{3}\approx113.04\text{ cm}^{3}\).