Question

section 11.7 homework

1. the shaded circle at right has area (40pi\text{ cm}^2). find the surface area of the sphere.

1. find the volume of a sphere whose surface area is (64pi\text{ cm}^2).

Explanation:

Identify the great circle area

The shaded circle represents a great circle of the sphere with radius \(r\).
Using the Great Circle Area knowledge point:

$$ A_{\text{great circle}} = \pi r^2 = 40\pi \implies r^2 = 40 $$

Calculate the surface area of the sphere

Using the Sphere Surface Area knowledge point:

$$ A_{\text{surface}} = 4\pi r^2 = 4(40\pi) = 160\pi \text{ cm}^2 $$

Find the radius from the given surface area

For Question 5, we are given the surface area of a sphere.
Using the Sphere Surface Area knowledge point:

$$ 4\pi r^2 = 64\pi \implies r^2 = 16 \implies r = 4 \text{ cm} $$

Calculate the volume of the sphere

Using the Sphere Volume knowledge point:

$$ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (4)^3 = \frac{256\pi}{3} \text{ cm}^3 $$

Answer:

Question 4

The surface area of the sphere is \(160\pi \text{ cm}^2\).

Question 5

The volume of the sphere is \(\frac{256\pi}{3} \text{ cm}^3\).