The volume of a rectangular block of stone is 10368 dm³. Its dimensions are in the ratio of 3 : 2 : 1. If its entire surface is polished at 2 paise per dm², then the total cost will be

### Concept & Volume and Surface Area Ratios To find the cost of polishing the surface, we must first determine the actual dimensions from the given ratio and volume, calculate the total surface area, and then multiply by the given rate. $$ Volume = 3x \times 2x \times 1x $$ $$ Surface\_Area = 2(lb + bh + hl) $$ ### Step-by-Step Solution 1. **Find the Actual Dimensions**: - Let the dimensions be 3x, 2x, and 1x. - Volume = 3x × 2x × x = 6x³ - We are given Volume = 10368 dm³. - 6x³ = 10368 => x³ = 1728 - Taking the cube root, x = 12. - Dimensions are: L = 36 dm, B = 24 dm, H = 12 dm. 2. **Calculate Total Surface Area (TSA)**: - TSA = 2(36×24 + 24×12 + 12×36) - TSA = 2(864 + 288 + 432) - TSA = 2(1584) = 3168 dm² 3. **Calculate the Total Cost**: - Cost per dm² = 2 paise - Total Cost = 3168 × 2 = 6336 paise - Convert paise to Rupees (100 paise = 1 Rupee): 6336 / 100 = ₹ 63.36 ### Exam Strategy & Shortcut Recognize that x³ = 1728 is a standard cube. Knowing cubes up to 15 is crucial for speed (12³ = 1728). After finding TSA = 3168, notice that the last digit calculation (8 × 2 = 16, ends in 6) immediately points to option (d) as it's the only one ending in a 6 in the hundredths place. ### Common Pitfall A very common mistake is forgetting to convert the final answer from paise to Rupees, leaving the student confused when 6336 doesn't perfectly match an option formatted as rupees. ### Final Answer Therefore, the correct answer is **₹ 63.36**.