From a cube of side 8m, a square hole of 3m side is hollowed from end to end. What is the volume of the remaining solid?

### Concept & Strategy When a hole is drilled through a solid, the volume of the remaining solid is the original volume minus the volume of the material removed. Since the hole is square and goes "from end to end," the removed material forms a cuboid. $$Remaining Volume = Volume_{cube} - Volume_{hole}$$ ### Step-by-Step Solution 1. **Calculate the Original Volume:** Side of the cube, $a = 8$ m. $Volume_{cube} = 8^3 = 8 \times 8 \times 8 = 512$ m$^3$. 2. **Calculate the Volume of the Hole:** The hole has a square cross-section of $3$ m $\times$ $3$ m. Because it goes from end to end of the 8m cube, its length (height) is $8$ m. $Volume_{hole} = L \times B \times H = 3 \times 3 \times 8 = 72$ m$^3$. 3. **Calculate Remaining Volume:** $Remaining Volume = 512 - 72 = 440$ m$^3$. ### Exam Strategy & Shortcut Visualize the removed piece as a prism with a base area of $3^2=9$. The volume of any uniform prism is $Base Area \times Length$. $9 \times 8 = 72$. Subtracting 72 from $8^3$ (512) takes only seconds. ### Common Pitfall Assuming the hole is a smaller cube (i.e., subtracting $3^3 = 27$ instead of $3 \times 3 \times 8 = 72$). The phrase "from end to end" indicates the hole spans the entire 8m depth of the original cube. ### Final Answer Therefore, the correct answer is **440 m$^3$**.