Question

You are given the great circle of a sphere is a length of 60 miles. What is the volume of the sphere?

Question 4 options:

6486.2 mi³


3591.4 mi³


4621.3 mi³


1764.7 mi³

Answer 1

To find the volume of the sphere given the length of a great circle, we first need to find the radius of the sphere. The circumference \( C \) of a sphere (which is the length of the great circle) is given by the formula:

\[
C = 2 \pi r
\]

where \( r \) is the radius of the sphere. Given that the circumference is 60 miles, we can set up the equation:

\[
60 = 2 \pi r
\]

To solve for the radius \( r \):

\[
r = \frac{60}{2 \pi} = \frac{30}{\pi}
\]

Next, we need to calculate the volume \( V \) of the sphere, which is given by the formula:

\[
V = \frac{4}{3} \pi r^3
\]

Now, substituting \( r \) into the volume formula:

\[
V = \frac{4}{3} \pi \left(\frac{30}{\pi}\right)^3
\]

Calculating \( \left(\frac{30}{\pi}\right)^3 \):

\[
\left(\frac{30}{\pi}\right)^3 = \frac{30^3}{\pi^3} = \frac{27000}{\pi^3}
\]

Now, substituting this back into the volume formula:

\[
V = \frac{4}{3} \pi \cdot \frac{27000}{\pi^3}
\]

This simplifies to:

\[
V = \frac{4 \cdot 27000}{3 \pi^2}
\]

Calculating \( \frac{4 \cdot 27000}{3} = 36000 \):

\[
V = \frac{36000}{\pi^2}
\]

Importantly, we can use an approximation for \( \pi \), say \( \pi \approx 3.14159 \):

\[
\pi^2 \approx 9.8696
\]

Now calculating \( \frac{36000}{9.8696} \):

\[
V \approx 3641.6 \text{ mi}^3
\]

Now let's review the options given:

1. 6486.2 mi³
2. 3591.4 mi³
3. 4621.3 mi³
4. 1764.7 mi³

The volume we calculated (approximately 3641.6 mi³) is closest to the option:

**3591.4 mi³**.

Thus, the answer is:

\[
\boxed{3591.4 \text{ mi}^3}
\]