How many lead shots each $3$ mm in diameter can be made from a cuboid of dimensions $9 \text{ cm} \times 11 \text{ cm} \times 12 \text{ cm}$?

### Concept & Conservation of Volume and Unit Conversion To find the number of identical small objects formed from melting a larger object, divide the volume of the larger object by the volume of a single small object. All measurements must be converted to the same unit before calculation. $$n = \frac{\text{Volume of Cuboid}}{\text{Volume of one Lead Shot}}$$ ### Step-by-Step Solution * **Given:** Cuboid dimensions $= 9 \text{ cm} \times 11 \text{ cm} \times 12 \text{ cm}$. * Volume of Cuboid $= 9 \times 11 \times 12 = 1188 \text{ cm}^3$. * Diameter of lead shot $= 3$ mm. Convert to cm: $3 \text{ mm} = 0.3$ cm. * Radius of lead shot $r = \frac{0.3}{2} = 0.15$ cm (or $\frac{3}{20}$ cm). * Volume of one spherical shot $= \frac{4}{3} \times \pi \times r^3$. * $= \frac{4}{3} \times \frac{22}{7} \times \left(\frac{3}{20}\right)^3 = \frac{4}{3} \times \frac{22}{7} \times \frac{27}{8000}$. * $= \frac{88}{21} \times \frac{27}{8000} = \frac{11}{7} \times \frac{9}{1000} = \frac{99}{7000} \text{ cm}^3$. * Number of shots $n = \frac{1188}{\frac{99}{7000}} = 1188 \times \frac{7000}{99}$. * Divide $1188$ by $99$: $1188 \div 99 = 12$. * $n = 12 \times 7000 = 84000$. ### Exam Strategy & Shortcut Work entirely in fractions and do not calculate intermediate decimal values. Set up the grand fraction: $N = \frac{9 \times 11 \times 12}{\frac{4}{3} \times \frac{22}{7} \times \frac{3}{20} \times \frac{3}{20} \times \frac{3}{20}}$. Notice how the $11$ in the numerator beautifully cancels the $22$ in the denominator, reducing the math burden drastically. ### Common Pitfall Failing to convert millimeters to centimeters. Mixing units will result in an answer off by a massive magnitude (like $72$ or $84$ instead of $84000$). ### Final Answer Therefore, the correct answer is **84000**.