A cone of height $7\text{ cm}$ and base radius $3\text{ cm}$ is carved from a rectangular block of wood $10\text{ cm} \times 5\text{ cm} \times 2\text{ cm}$. The percentage of wood wasted is

### Concept & Percentage Waste The wood wasted is the original volume of the block minus the volume of the carved cone. The percentage of waste is the ratio of this wasted volume to the original volume, multiplied by 100. $$Volume\ of\ Cuboid = l \times w \times h$$ $$Volume\ of\ Cone = \frac{1}{3}\pi r^2 h$$ $$Percentage\ Wasted = \left(\frac{Original\ Volume - Cone\ Volume}{Original\ Volume}\right) \times 100$$ ### Step-by-Step Solution * **Step 1:** Calculate the volume of the rectangular block (cuboid). $V_{block} = 10\text{ cm} \times 5\text{ cm} \times 2\text{ cm} = 100\text{ cm}^3$ * **Step 2:** Calculate the volume of the carved cone. $V_{cone} = \frac{1}{3} \times \pi \times r^2 \times h$ $V_{cone} = \frac{1}{3} \times \frac{22}{7} \times (3)^2 \times 7$ $V_{cone} = \frac{1}{3} \times \frac{22}{7} \times 9 \times 7 = 22 \times 3 = 66\text{ cm}^3$ * **Step 3:** Determine the volume of wood wasted. $Waste = V_{block} - V_{cone} = 100 - 66 = 34\text{ cm}^3$ * **Step 4:** Calculate the percentage of waste. $Percentage = \left(\frac{34}{100}\right) \times 100 = 34\%$ ### Exam Strategy & Shortcut Notice that the rectangular block volume is precisely $100$. This makes the percentage calculation trivial, as the volume wasted numerically equals the percentage wasted. Solve for the cone volume ($66$) and mentally subtract from $100$ to immediately get $34\%$. ### Common Pitfall A common mistake is finding the volume of the cone ($66\%$) and selecting that as the answer, forgetting that the question specifically asks for the percentage of wood *wasted*, not the percentage utilized. ### Final Answer Therefore, the correct answer is **34%**.