A cone of height 15 cm and base diameter 30 cm is carved out of a wooden sphere of radius 15 cm. The percentage of wood wasted is

### Concept & Percentage Difference To find the percentage of material wasted when carving an object, subtract the volume of the carved object from the initial volume, then divide by the initial volume and multiply by 100. $$\text{Wasted %} = \left( \frac{V_{initial} - V_{carved}}{V_{initial}} \right) \times 100$$ ### Step-by-Step Solution 1. **Initial Volume (Sphere):** Radius ($R$) = $15\text{ cm}$. $V_{sphere} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (15)^3 = 4500\pi\text{ cm}^3$ 2. **Carved Volume (Cone):** Height ($h$) = $15\text{ cm}$, Base diameter = $30\text{ cm} \Rightarrow$ radius ($r$) = $15\text{ cm}$. $V_{cone} = \frac{1}{3}\pi r^2h = \frac{1}{3}\pi (15)^2(15) = \frac{1}{3}\pi (15)^3 = 1125\pi\text{ cm}^3$ 3. **Calculate Wasted Wood:** Wasted Volume = $V_{sphere} - V_{cone} = 4500\pi - 1125\pi = 3375\pi\text{ cm}^3$ 4. **Calculate Percentage:** $\text{Percentage Wasted} = \left( \frac{3375\pi}{4500\pi} \right) \times 100 = \left( \frac{3375}{4500} \right) \times 100$ $\text{Percentage Wasted} = \left( \frac{3}{4} \right) \times 100 = 75\%$ ### Exam Strategy & Shortcut Instead of calculating the large actual volumes, evaluate their ratios directly. $V_{sphere} = \frac{4}{3}\pi (15)^3$ $V_{cone} = \frac{1}{3}\pi (15)^3$ Ratio of $V_{cone}$ to $V_{sphere} = \frac{1}{4}$ (or $25\%$). This means $25\%$ of the wood is used for the cone, so $100\% - 25\% = 75\%$ is wasted. ### Common Pitfall Using the diameter ($30\text{ cm}$) instead of the radius ($15\text{ cm}$) to calculate the cone's volume, resulting in a completely incorrect waste percentage. ### Final Answer Therefore, the correct answer is **75%**.