A boat has to travel upstream 20 km distance from point X of a river to point Y. The total time taken by boat in travelling from point X to Y and Y to X is 41 min 40 s. What is the speed of the boat?

Options

A. a. 66 km/h
B. b. 72 km/h
C. c. 48 km/h
D. d. Cannot be determined

Answer

To solve the problem step by step, we need to analyze the situation involving the boat traveling upstream and downstream in a river.

### Step 1: Understand the problem
The boat travels a distance of 20 km upstream from point X to point Y and then returns downstream from Y to X. The total time taken for this round trip is 41 minutes and 40 seconds.

### Step 2: Convert the total time into hours
Since speed is usually expressed in km/h, we need to convert the total time taken into hours.

- Total time = 41 minutes and 40 seconds
- Convert minutes to hours:
\[ 41 \text{ minutes} = \frac{41}{60} \text{ hours} \approx 0.6833 \text{ hours} \]
- Convert seconds to hours:
\[ 40 \text{ seconds} = \frac{40}{3600} \text{ hours} \approx 0.0111 \text{ hours} \]
- Total time in hours:
\[ \text{Total time} = 0.6833 + 0.0111 \approx 0.6944 \text{ hours} \]

### Step 3: Set up the equations for speed
Let:
- \( A \) = speed of the boat in still water (km/h)
- \( B \) = speed of the river current (km/h)

When the boat travels upstream (against the current), its effective speed is \( A - B \). When it travels downstream (with the current), its effective speed is \( A + B \).

### Step 4: Write the time equations
The time taken to travel upstream (20 km) is:
\[ \text{Time upstream} = \frac{20}{A - B} \]

The time taken to travel downstream (20 km) is:
\[ \text{Time downstream} = \frac{20}{A + B} \]

The total time for the round trip is:
\[ \frac{20}{A - B} + \frac{20}{A + B} = 0.6944 \]

### Step 5: Simplify the equation
Multiply through by \( (A - B)(A + B) \) to eliminate the denominators:
\[ 20(A + B) + 20(A - B) = 0.6944(A^2 - B^2) \]
This simplifies to:
\[ 40A = 0.6944(A^2 - B^2) \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 0.6944A^2 - 40A - 0.6944B^2 = 0 \]

### Step 7: Analyze the equation
This is a quadratic equation in terms of \( A \). However, we have two unknowns \( A \) and \( B \) and only one equation, which means we cannot uniquely determine \( A \) without additional information about \( B \).

### Conclusion
Since we cannot determine the speed of the boat without knowing the speed of the river, the problem does not have a unique solution.