To solve this problem, let's break it down step-by-step:
1. Calculate the volume of the rectangular prism:
- The base of the prism is a square with sides measuring 5 cm.
- The height of the prism is 12 cm.
- The volume [tex]\(V_{\text{prism}}\)[/tex] of a rectangular prism is given by the formula:
[tex]\[
V_{\text{prism}} = \text{base area} \times \text{height}
\][/tex]
Since the base is a square:
[tex]\[
\text{base area} = \text{side}^2 = 5 \times 5 = 25 \, \text{cm}^2
\][/tex]
Therefore:
[tex]\[
V_{\text{prism}} = 25 \, \text{cm}^2 \times 12 \, \text{cm} = 300 \, \text{cm}^3
\][/tex]
2. Calculate the volume of the pyramid:
- The pyramid has the same base as the prism, which is a square with side 5 cm.
- The height of the pyramid is half the height of the prism, which is:
[tex]\[
\text{height of pyramid} = \frac{12}{2} = 6 \, \text{cm}
\][/tex]
- The volume [tex]\(V_{\text{pyramid}}\)[/tex] of a pyramid is given by the formula:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{base area} \times \text{height}
\][/tex]
Using the previously calculated base area of 25 cm²:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times 25 \, \text{cm}^2 \times 6 \, \text{cm}
\][/tex]
Simplifying:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times 150 \, \text{cm}^3 = 50 \, \text{cm}^3
\][/tex]
3. Calculate the volume of the space outside the pyramid but inside the prism:
- To find this volume, subtract the volume of the pyramid from the volume of the prism:
[tex]\[
\text{space volume} = V_{\text{prism}} - V_{\text{pyramid}}
\][/tex]
Using the volumes calculated above:
[tex]\[
\text{space volume} = 300 \, \text{cm}^3 - 50 \, \text{cm}^3 = 250 \, \text{cm}^3
\][/tex]
The volume of the space outside the pyramid but inside the prism is:
[tex]\[
250 \, \text{cm}^3
\][/tex]
Therefore, the correct choice is:
D. 250 cm³
