Question

determine the volume and the surface area of the three-dimensional figure.
use the \\(\pi\\) key on your calculator.

a) the volume is \\(113.10\\) \\(\text{cm}^3\\).
(simplify your answer. type an integer or a decimal rounded to the nearest hundredth as needed.)

b) the surface area is \\(144.85\\) \\(\text{cm}^2\\).
(simplify your answer. type an integer or a decimal rounded to the nearest hundredth as needed.)

Explanation:

Identify the given dimensions

The given figure is a cone with:

• Radius \(r = 3\text{ cm}\)
• Height \(h = 12\text{ cm}\)

Calculate the volume of the cone

Using the Volume of Cone knowledge point

$$ V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (3)^2 (12) = 36\pi \approx 113.0973\text{ cm}^3 $$

Rounded to the nearest hundredth, the volume is \(113.10\text{ cm}^3\).

Find the slant height of the cone

Using the Pythagorean Theorem knowledge point

$$ l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 12^2} = \sqrt{9 + 144} = \sqrt{153} \approx 12.3693\text{ cm} $$

Calculate the surface area of the cone

Using the Surface Area of Cone knowledge point

$$ A = \pi r^2 + \pi r l = \pi (3)^2 + \pi (3)(\sqrt{153}) = 9\pi + 3\sqrt{153}\pi \approx 28.2743 + 116.5786 = 144.853\text{ cm}^2 $$

Rounded to the nearest hundredth, the surface area is \(144.85\text{ cm}^2\).

Answer:

Question 1

a) The volume is <blank>\(113.10\)</blank> \(\text{cm}^3\).

Question 2

b) The surface area is <blank>\(144.85\)</blank> \(\text{cm}^2\).