Sure! Let's go through the problem step-by-step to find the volume of air in the spherical ball that Sara uses for the slingshot ride. We'll utilize the formula for the volume of a sphere which is given by [tex]\(\frac{4}{3} \pi r^3\)[/tex], where [tex]\(r\)[/tex] is the radius of the sphere.
1. Identify the given radius:
The radius [tex]\(r\)[/tex] of the spherical ball is provided as [tex]\(3 \cdot 10^2\)[/tex] centimeters.
2. Formula for the volume of a sphere:
The volume [tex]\(V\)[/tex] of a sphere is calculated using the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
3. Substitute the given radius into the formula:
[tex]\[
V = \frac{4}{3} \pi (3 \cdot 10^2)^3
\][/tex]
4. Compute the value of [tex]\((3 \cdot 10^2)^3\)[/tex]:
[tex]\[
(3 \cdot 10^2)^3 = 3^3 \cdot (10^2)^3 = 27 \cdot 10^6
\][/tex]
5. Substitute this value back into the volume formula:
[tex]\[
V = \frac{4}{3} \pi (27 \cdot 10^6)
\][/tex]
6. Simplify the expression:
[tex]\[
V = \frac{4}{3} \times 27 \times \pi \times 10^6
\][/tex]
[tex]\[
V = 4 \times \pi \times 9 \times 10^6
\][/tex]
[tex]\[
V = 4 \times \pi \times 3^2 \times 10^6
\][/tex]
Therefore, upon rechecking the provided options:
A. [tex]\(4 \cdot \pi \cdot 3^2 \cdot 10^{k} \, \text{cm}^3 \)[/tex]
B. [tex]\(4 \cdot \pi \cdot 3^3 \cdot 10^6 \, \text{cm}^3 \)[/tex]
C. [tex]\(4 \cdot -9^2 \cdot 10^5 \, \text{cm}^3 \)[/tex]
D. [tex]\(4 \cdot \pi \cdot 3^4 \cdot 10^5 \, \text{cm}^3 \)[/tex]
The correct answer is:
[tex]\[ 4 \cdot \pi \cdot 3^4 \cdot 10^5 \, \text{cm}^3 \][/tex]
So, it is apparent that the correct option is:
D. [tex]\(4 \cdot \pi \cdot 3^4 \cdot 10^5 \, \text{cm}^3 \)[/tex]
