The cost of painting the whole surface area of a cube at the rate of 13 paise per sq. cm is ₹ 343.98. Then the volume of the cube is :

### Concept & Strategy The total cost of painting is the product of the total surface area and the rate per square unit. By dividing the total cost by the rate, we can find the surface area. Then, deduce the side length and compute the volume. $$Total Surface Area = \frac{Total Cost}{Rate}$$ $$Surface Area = 6a^2$$ $$Volume = a^3$$ ### Step-by-Step Solution 1. **Convert Units:** Total Cost = ₹ $343.98 = 34398$ paise. Rate = $13$ paise per sq. cm. 2. **Find Surface Area:** $Surface Area = \frac{34398}{13}$ $Surface Area = 2646$ cm$^2$. 3. **Find Side Length ($a$):** $6a^2 = 2646$ $a^2 = \frac{2646}{6} = 441$ $a = \sqrt{441} = 21$ cm. 4. **Calculate Volume:** $Volume = 21^3$ $Volume = 21 \times 21 \times 21 = 9261$ cm$^3$. ### Exam Strategy & Shortcut Once you determine that $a = 21$, you need to find $21^3$. The unit digit of $21^3$ is $1 \times 1 \times 1 = 1$. Looking at the options, only $9261$ ends in 1. You can confidently select this without calculating $21 \times 21 \times 21$. ### Common Pitfall Failing to align units. The rate is given in paise, but the total cost is in rupees. If you don't convert ₹ $343.98$ to $34398$ paise (or 13 paise to ₹ 0.13), your surface area calculation will be off by a factor of 100. ### Final Answer Therefore, the correct answer is **9261 cm$^3$**.