A certain number of persons can dig a trench 100 m long, 50 m broad and 10 m deep in 10 days. The same number of persons can dig another trench 20 m broad and 15 m deep in 30 days. The length of the second trench is

### Concept & Work and Volume The volume of the trench represents the total work done ($W = \text{Length} \times \text{Breadth} \times \text{Depth}$). Since the number of persons ($M$) is the same in both cases, we relate work and days ($D$): $$ \frac{M \times D_1}{W_1} = \frac{M \times D_2}{W_2} \implies \frac{D_1}{W_1} = \frac{D_2}{W_2} $$ ### Step-by-Step Solution * **Given**: - Trench 1: $L_1 = 100$ m, $B_1 = 50$ m, $D_1^{depth} = 10$ m, $D_1^{time} = 10$ days. - Trench 2: $B_2 = 20$ m, $D_2^{depth} = 15$ m, $D_2^{time} = 30$ days. Let length be $L_2$. * **Calculation**: - $W_1 = 100 \times 50 \times 10 = 50,000 \, \text{m}^3$ - $W_2 = L_2 \times 20 \times 15 = 300 L_2 \, \text{m}^3$ Using the formula $\frac{D_1}{W_1} = \frac{D_2}{W_2}$: $$ \frac{10}{50000} = \frac{30}{300 L_2} $$ Simplify fractions: $$ \frac{1}{5000} = \frac{1}{10 L_2} $$ $$ 10 L_2 = 5000 \implies L_2 = 500 \, \text{m}?? \text{ Wait, let's re-verify calculation.} $$ Let's check carefully: $$ \frac{10}{50000} = \frac{30}{300 L_2} \implies \frac{1}{5000} = \frac{1}{10 L_2} \implies 10 L_2 = 5000 \implies L_2 = 500. $$ Wait, let's look at the options and standard solution for this classic problem. Let's search or re-calculate. Let's re-evaluate: $W_1 = 100 \times 50 \times 10 = 50,000$. $W_2 = L_2 \times 20 \times 15 = 300 L_2$. Formula: $\frac{M_1 \cdot D_1}{W_1} = \frac{M_2 \cdot D_2}{W_2}$ Here $M_1 = M_2$, so: $\frac{10}{50000} = \frac{30}{300 L_2}$ Cross multiply: $10 \times (300 L_2) = 30 \times 50000$ $3000 L_2 = 1,500,000$ $L_2 = \frac{1,500,000}{3000} = 500$. Wait, option (b) is $500$ m! Let me check option b. Option (b) is 500 m. Let's output 500 m. ### Exam Strategy & Shortcut Express total work as volume and set up the direct proportionality between days and volume when the workforce is identical. ### Common Pitfall Multiplying dimensions incorrectly or mixing up length, breadth, and depth positions in the volume product. ### Final Answer Therefore, the correct answer is **500 m**.