If 600 men dig a 5.5 m wide, 4 m deep and 405 m long canal in half an hour, then how long a canal will 2500 men, working for 6 hours, dig if it is 10 m wide and 8 m deep?

### Concept & Volumetric Work in Chain Rule In this variation of the chain rule, the "Work" ($W$) is the physical volume of the canal dug. Volume is calculated as Length $\times$ Width $\times$ Depth. We substitute this product into the denominator of the standard formula. $$ \frac{M_1 H_1}{V_1} = \frac{M_2 H_2}{V_2} $$ ### Step-by-Step Solution * Establish Scenario 1 variables: $M_1 = 600 \text{ men}$, $H_1 = 0.5 \text{ hours}$, $V_1 = 5.5 \times 4 \times 405 = 8910 \text{ m}^3$. * Establish Scenario 2 variables: $M_2 = 2500 \text{ men}$, $H_2 = 6 \text{ hours}$, $V_2 = 10 \times 8 \times L = 80L \text{ m}^3$ (where $L$ is the unknown length). * Apply the chain rule equation: $$ \frac{600 \times 0.5}{5.5 \times 4 \times 405} = \frac{2500 \times 6}{10 \times 8 \times L} $$ * Simplify the numerators: $$ \frac{300}{8910} = \frac{15000}{80L} $$ * Simplify the fractions by cancelling zeros: $$ \frac{30}{891} = \frac{1500}{8L} $$ * Cross-multiply to isolate $L$: $$ 30 \times 8L = 1500 \times 891 $$ $$ 240L = 1336500 $$ * Solve for $L$: $$ L = \frac{1336500}{240} = \frac{133650}{24} $$ * Divide to get the final decimal/fraction: $133650 \div 24 = 5568.75$, which equals $5568 \frac{3}{4} \text{ m}$. ### Exam Strategy & Shortcut To prevent massive multiplication, keep numbers factored and cancel them across the equals sign. From $\frac{600 \times 0.5}{5.5 \times 4 \times 405} = \frac{2500 \times 6}{10 \times 8 \times L}$, immediately cancel the $600 \times 0.5$ ($300$) with the $2500 \times 6$ ($15000$) to leave a $50$ on the right side. The equation shrinks rapidly to $\frac{1}{5.5 \times 4 \times 405} = \frac{50}{80L}$. This heavily reduces the arithmetic burden. ### Common Pitfall A major pitfall is converting "half an hour" into 30 minutes for $H_1$, but keeping $H_2$ as 6 hours. Units of time must be strictly consistent (both in hours or both in minutes). ### Final Answer Therefore, the correct answer is **$5568 \frac{3}{4}$ m**.