The diameter of a sphere is 8 cm. It is melted and drawn into a wire of diameter 3 mm. The length of the wire is

### Concept & Volume Conservation Melting and reshaping a solid means equating the original volume (sphere) with the new volume (cylinder/wire). $$Volume_{sphere} = \frac{4}{3}\pi R^3$$ $$Volume_{wire} = \pi r^2 h$$ ### Step-by-Step Solution 1. Given sphere diameter = $8 \text{ cm}$, so its radius $R = 4 \text{ cm}$. 2. Volume of the sphere = $\frac{4}{3}\pi (4)^3 = \frac{256}{3}\pi \text{ cm}^3$. 3. Given wire diameter = $3 \text{ mm} = 0.3 \text{ cm}$. Its radius $r = 0.15 \text{ cm} = \frac{3}{20} \text{ cm}$. 4. Volume of the wire = $\pi (\frac{3}{20})^2 \times h = \frac{9}{400}\pi h \text{ cm}^3$. 5. Equate the volumes: $\frac{9}{400}\pi h = \frac{256}{3}\pi$. 6. Cancel $\pi$ and solve for $h$: $h = \frac{256 \times 400}{9 \times 3} = \frac{102400}{27} \text{ cm}$. 7. Calculate the value: $h \approx 3792.59 \text{ cm}$. 8. Convert to meters by dividing by 100: $37.9259 \text{ m} \approx 37.9 \text{ m}$. ### Exam Strategy & Shortcut Instead of working with the decimal $0.15$ for the wire's radius, use the fraction $\frac{3}{20}$. Fractions are much easier to square and cross-multiply in volume conservation problems. ### Common Pitfall Failing to convert the final answer from centimeters to meters. The raw calculation yields $\approx 3792$, which might cause confusion if the units in the options are not carefully checked. ### Final Answer Therefore, the correct answer is **37.9 m**.