The problem asks to complete a table for spheres given some information such as radius, surface area, or volume. The answers need to be expressed in terms of $\pi$. We need to find the circumference of the great circle, surface area, and volume for different spheres given their radius.

1. Problem Description

The problem asks to complete a table for spheres given some information such as radius, surface area, or volume. The answers need to be expressed in terms of $\pi$. We need to find the circumference of the great circle, surface area, and volume for different spheres given their radius.

2. Solution Steps

For problem 11:

Given radius $r = 7$ mm.

Circumference of the great circle: $C = 2 \pi r = 2 \pi (7) = 14 \pi$ mm.

Surface area of the sphere: $S = 4 \pi r^2 = 4 \pi (7^2) = 4 \pi (49) = 196 \pi$ mm$^2$.

Volume of the sphere: $V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (7^3) = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi$ mm$^3$.

For problem 12:

Given surface area $S = 144 \pi$ in$^2$.

We know $S = 4 \pi r^2$.

So, $144 \pi = 4 \pi r^2$.

$r^2 = \frac{144 \pi}{4 \pi} = 36$.

$r = \sqrt{36} = 6$ in.

Circumference of the great circle: $C = 2 \pi r = 2 \pi (6) = 12 \pi$ in.

Volume of the sphere: $V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (6^3) = \frac{4}{3} \pi (216) = 4 \pi (72) = 288 \pi$ in$^3$.

For problem 13:

Given the circumference of the great circle $C = 10 \pi$ cm.

We know $C = 2 \pi r$.

So, $10 \pi = 2 \pi r$.

$r = \frac{10 \pi}{2 \pi} = 5$ cm.

Surface area of the sphere: $S = 4 \pi r^2 = 4 \pi (5^2) = 4 \pi (25) = 100 \pi$ cm$^2$.

Volume of the sphere: $V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (5^3) = \frac{4}{3} \pi (125) = \frac{500}{3} \pi$ cm$^3$.

For problem 14:

Given the volume $V = \frac{4000 \pi}{3}$ m$^3$.

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

So, $\frac{4000 \pi}{3} = \frac{4}{3} \pi r^3$.

$r^3 = \frac{4000 \pi / 3}{4 \pi / 3} = \frac{4000 \pi}{3} \times \frac{3}{4 \pi} = 1000$.

$r = \sqrt[3]{1000} = 10$ m.

Circumference of the great circle: $C = 2 \pi r = 2 \pi (10) = 20 \pi$ m.

Surface area of the sphere: $S = 4 \pi r^2 = 4 \pi (10^2) = 4 \pi (100) = 400 \pi$ m$^2$.

3. Final Answer

Problem 11:

Circumference of the great circle: $14 \pi$ mm

Surface area of the sphere: $196 \pi$ mm$^2$

Volume of the sphere: $\frac{1372}{3} \pi$ mm$^3$

Problem 12:

Radius: 6 in

Circumference of the great circle: $12 \pi$ in

Volume of the sphere: $288 \pi$ in$^3$

Problem 13:

Radius: 5 cm

Surface area of the sphere: $100 \pi$ cm$^2$

Volume of the sphere: $\frac{500}{3} \pi$ cm$^3$

Problem 14:

Radius: 10 m

Circumference of the great circle: $20 \pi$ m

Surface area of the sphere: $400 \pi$ m$^2$