The speed of the boat in still water in 15 km/hr. If the boat travels 54 km each in downstream and upstream in 7.5 hr, then find the time taken by the boat to travel 48 km in upstream?

Options

A. 8 hrs
B. 6 hrs
C. 4 hrs
D. 5 hrs

Answer

To solve the problem, we need to find the time taken by the boat to travel 48 km upstream. We know the speed of the boat in still water and the total time taken for both downstream and upstream trips. Let's break it down step by step.

### Step 1: Define Variables
Let:
- Speed of the boat in still water (B) = 15 km/hr
- Speed of the stream (S)
- Speed of the boat downstream (B + S)
- Speed of the boat upstream (B - S)

### Step 2: Set Up the Equations
The boat travels 54 km downstream and 54 km upstream in a total time of 7.5 hours. We can express the time taken for each part of the journey as follows:

- Time taken downstream = Distance / Speed = 54 / (B + S)
- Time taken upstream = Distance / Speed = 54 / (B - S)

Thus, we have the equation:
\[ \frac{54}{B + S} + \frac{54}{B - S} = 7.5 \]

### Step 3: Substitute the Known Value
Substituting B = 15 km/hr into the equation:
\[ \frac{54}{15 + S} + \frac{54}{15 - S} = 7.5 \]

### Step 4: Solve for S
To solve for S, we first find a common denominator:
\[ \frac{54(15 - S) + 54(15 + S)}{(15 + S)(15 - S)} = 7.5 \]
This simplifies to:
\[ \frac{54(30)}{(15 + S)(15 - S)} = 7.5 \]
\[ \frac{1620}{(15 + S)(15 - S)} = 7.5 \]
Cross-multiplying gives:
\[ 1620 = 7.5(15^2 - S^2) \]
\[ 1620 = 7.5(225 - S^2) \]
Dividing both sides by 7.5:
\[ 216 = 225 - S^2 \]
\[ S^2 = 225 - 216 \]
\[ S^2 = 9 \]
\[ S = 3 \text{ km/hr} \]

### Step 5: Calculate Speeds
Now we can find the speeds:
- Speed downstream = B + S = 15 + 3 = 18 km/hr
- Speed upstream = B - S = 15 - 3 = 12 km/hr

### Step 6: Find Time for 48 km Upstream
To find the time taken to travel 48 km upstream:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{48}{12} = 4 \text{ hours} \]

### Final Answer
The time taken by the boat to travel 48 km upstream is **4 hours**.

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