To solve this problem, we need to calculate the volumes of the cube and the sphere, then find the volume of the space in the cube not taken up by the sphere.
### Step 1: Calculate the volume of the cube
The volume [tex]\( V \)[/tex] of a cube is given by the formula:
[tex]\[ V_{\text{cube}} = \text{edge length}^3 \][/tex]
Given:
- Edge length of the cube = 60 cm
So,
[tex]\[ V_{\text{cube}} = 60^3 \][/tex]
[tex]\[ V_{\text{cube}} = 60 \times 60 \times 60 \][/tex]
[tex]\[ V_{\text{cube}} = 216,000 \text{ cubic centimeters} \][/tex]
### Step 2: Calculate the volume of the sphere
The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Given:
- Radius of the sphere ([tex]\( r \)[/tex]) = 30 cm
- [tex]\( \pi \approx 3.141592653589793 \)[/tex]
So,
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi (30)^3 \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi \times 27,000 \][/tex]
[tex]\[ V_{\text{sphere}} \approx \frac{4}{3} \times 3.141592653589793 \times 27,000 \][/tex]
[tex]\[ V_{\text{sphere}} \approx 113,097 \text{ cubic centimeters} \][/tex]
### Step 3: Calculate the volume of the space in the cube not taken up by the sphere
We subtract the volume of the sphere from the volume of the cube:
[tex]\[ V_{\text{remaining}} = V_{\text{cube}} - V_{\text{sphere}} \][/tex]
So,
[tex]\[ V_{\text{remaining}} = 216,000 - 113,097 \][/tex]
[tex]\[ V_{\text{remaining}} \approx 102,903 \text{ cubic centimeters} \][/tex]
### Conclusion:
The volume of the space in the cube that is not taken up by the sphere, to the nearest cubic centimeter, is approximately [tex]\( 102,903 \)[/tex] cubic centimeters.
Thus, the correct answer is:
A) 102,903
