To determine the volume of a ball (or sphere) that is 24 cm across, Sheena needs to use the formula for the volume of a sphere. The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Here, [tex]\( r \)[/tex] is the radius of the sphere. Since the sphere is 24 cm across, the diameter of the sphere is 24 cm. The radius [tex]\( r \)[/tex] is half of the diameter. Thus:
[tex]\[ r = \frac{24}{2} = 12 \ \text{cm} \][/tex]
Now, substitute the radius into the volume formula:
[tex]\[ V = \frac{4}{3} \pi (12)^3 \][/tex]
Sheena's setup for calculating the volume of the ball should therefore be:
[tex]\[ V = \frac{4}{3} \pi (12)^3 \][/tex]
Let's take a look at the options provided:
1. [tex]\( V = \frac{1}{3} \pi 24^2(12) \)[/tex]
2. [tex]\( V = \frac{1}{3} \pi 12^2(24) \)[/tex]
3. [tex]\( V = \frac{4}{3} \pi 24^3 \)[/tex]
4. [tex]\( V = \frac{4}{3} \pi 12^3 \)[/tex]
Since we derived the correct equation to calculate the volume of the sphere as:
[tex]\[ V = \frac{4}{3} \pi (12)^3 \][/tex]
The correct option is:
[tex]\[ V = \frac{4}{3} \pi 12^3 \][/tex]
Hence, given the problem, Sheena should set up her equation as option 4: [tex]\( V = \frac{4}{3} \pi 12^3 \)[/tex].
Upon calculation, this yields the volume of the sphere:
[tex]\[ V \approx 7238.23 \ \text{cubic centimeters} \][/tex]
