An architectural model includes a rectangular prism (5 m by 3 m by 2 m) with a cylindrical hole (radius 0.75 m, depth 2 m) running through it. What is the remaining volume of the prism?

Sure, let's solve this step by step.

1. Calculate the volume of the rectangular prism: The formula for the volume of a rectangular prism is: $V_{\text{prism}} = \text{length} \times \text{width} \times \text{height}$ Given the dimensions: $V_{\text{prism}} = 5 \, \text{m} \times 3 \, \text{m} \times 2 \, \text{m} = 30 \, \text{m}^3$

2. Calculate the volume of the cylindrical hole: The formula for the volume of a cylinder is: $V_{\text{cylinder}} = \pi \times \text{radius}^2 \times \text{height}$ Given the radius and depth (height) of the cylinder: $V_{\text{cylinder}} = \pi \times (0.75 \, \text{m})^2 \times 2 \, \text{m}$ First, calculate the radius squared: $(0.75 \, \text{m})^2 = 0.5625 \, \text{m}^2$ Then, multiply by $\pi$ and the height: $V_{\text{cylinder}} = \pi \times 0.5625 \, \text{m}^2 \times 2 \, \text{m} = \pi \times 1.125 \, \text{m}^3$ Using $\pi \approx 3.14159$: $V_{\text{cylinder}} \approx 3.14159 \times 1.125 \, \text{m}^3 \approx 3.53429 \, \text{m}^3$

3. Calculate the remaining volume of the prism: Subtract the volume of the cylindrical hole from the volume of the rectangular prism: $V_{\text{remaining}} = V_{\text{prism}} - V_{\text{cylinder}}$ $V_{\text{remaining}} = 30 \, \text{m}^3 - 3.53429 \, \text{m}^3 \approx 26.46571 \, \text{m}^3$

So, the remaining volume of the prism is approximately $26.46571 \, \text{m}^3$.