A hollow spherical metallic ball has an external diameter $6$ cm and is $\frac{1}{2}$ cm thick. The volume of metal used in the ball is

### Concept & Volume of a Hollow Sphere The volume of material in a hollow sphere is the difference between the external volume and the internal volume. $$V = \frac{4}{3}\pi (R^3 - r^3)$$ Where $R$ is the external radius and $r$ is the internal radius. Internal radius $r = R - \text{thickness}$. ### Step-by-Step Solution * **Given:** External diameter $= 6$ cm $\Rightarrow$ External radius $R = 3$ cm. * Thickness $= 0.5$ cm. * Internal radius $r = 3 - 0.5 = 2.5$ cm $= \frac{5}{2}$ cm. * Volume of metal $= \frac{4}{3}\pi (R^3 - r^3)$ * $= \frac{4}{3} \times \frac{22}{7} \times \left(3^3 - \left(\frac{5}{2}\right)^3\right)$ * $= \frac{4}{3} \times \frac{22}{7} \times \left(27 - \frac{125}{8}\right)$ * $= \frac{4}{3} \times \frac{22}{7} \times \left(\frac{216 - 125}{8}\right)$ * $= \frac{4}{3} \times \frac{22}{7} \times \frac{91}{8}$ * Simplify the expression: $91 \div 7 = 13$, and $4 \div 8 = \frac{1}{2}$. * Volume $= \frac{1}{3} \times 22 \times 13 \times \frac{1}{2} = \frac{11 \times 13}{3} = \frac{143}{3}$ * Converting to a mixed fraction: $143 \div 3 = 47$ with a remainder of $2$. * Volume $= 47 \frac{2}{3} \text{ cm}^3$. ### Exam Strategy & Shortcut Keep terms in fraction format (like $5/2$) rather than decimals (like $2.5$) when substituting into volume equations. Cubing and subtracting fractions is structurally easier to simplify with factors like $22/7$. ### Common Pitfall Using the diameter ($6$ cm) directly in the formula instead of the radius ($3$ cm), or subtracting the thickness from the diameter instead of the radius to find the inner dimension. ### Final Answer Therefore, the correct answer is **$47 \frac{2}{3} \text{ cm}^3$**.