Question

section 11.7 homework

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

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

Explanation:

Question 4

Relate great circle area to radius

The shaded region represents a great circle of the sphere.
Using the Sphere Surface Area knowledge point:

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

Calculate the sphere surface area

The surface area formula of a sphere is:

$$ A_{\text{sphere}} = 4\pi r^2 $$

Substitute \(r^2 = 40\):

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

Question 5

Find the radius from surface area

Using the Sphere Surface Area knowledge point:

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

Calculate the sphere volume

Using the Sphere Volume knowledge point:

$$ V = \frac{4}{3}\pi r^3 $$

Substitute \(r = 4\):

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

Answer:

Question 4

\(160\pi \text{ cm}^2\)

Question 5

\(\frac{256}{3}\pi \text{ cm}^3\)