To determine the volume of the spherical ball in which Sara is strapped, we use the volume formula for a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Here, [tex]\( r \)[/tex] is the radius of the sphere.
Given:
[tex]\[ r = 3 \cdot 10^2 \ \text{cm} \][/tex]
Now, substitute [tex]\( r \)[/tex] into the volume formula:
[tex]\[ V = \frac{4}{3} \pi (3 \cdot 10^2)^3 \][/tex]
Calculate the term inside the parentheses first:
[tex]\[ (3 \cdot 10^2)^3 = 3^3 \cdot (10^2)^3 \][/tex]
[tex]\[ 3^3 = 27 \][/tex]
[tex]\[ (10^2)^3 = 10^{6} \][/tex]
So,
[tex]\[ (3 \cdot 10^2)^3 = 27 \cdot 10^6 \][/tex]
Next, substitute this back into the formula for volume:
[tex]\[ V = \frac{4}{3} \pi \cdot 27 \cdot 10^6 \][/tex]
[tex]\[ V = \frac{4 \cdot 27 \cdot \pi \cdot 10^6}{3} \][/tex]
[tex]\[ V = \frac{108 \cdot \pi \cdot 10^6}{3} \][/tex]
[tex]\[ V = 36 \cdot \pi \cdot 10^6 \][/tex]
We can now match this result with the provided options. The correct option is:
[tex]\[
B. \ 4 \cdot \pi \cdot 3^3 \cdot 10^6 \ \text{cm}^3
\][/tex]
Thus, the volume of the air in the spherical ball is given by option B:
[tex]\[
4 \cdot \pi \cdot 3^3 \cdot 10^6 \ \text{cm}^3
\][/tex]
