Two ghats are located on a river bank and are 21 km apart. Leaving one of the ghats for the other, a motorboat returns to the first ghat in 270 minutes, spending 40 minutes of that time in taking the passengers at the second ghat. Find the speed of the boat in still water if the speed of the river flow is 2.5 km/h?

Answer

To solve the problem, we need to find the speed of the boat in still water. Let's break down the solution step by step.

### Step 1: Understand the Given Information
- Distance between the two ghats = 21 km
- Total time for the round trip = 270 minutes
- Time spent at the second ghat = 40 minutes
- Speed of the river flow = 2.5 km/h

### Step 2: Convert Time into Hours
Since speed is usually expressed in km/h, we need to convert the total time from minutes to hours.
- Total time in hours = 270 minutes / 60 = 4.5 hours
- Time spent traveling = Total time - Time spent at the second ghat
- Time spent traveling = 4.5 hours - (40 minutes / 60) = 4.5 - (2/3) = 4.5 - 0.67 = 3.83 hours (approximately)

### Step 3: Determine the Effective Speeds
Let the speed of the boat in still water be \( x \) km/h.
- Speed of the boat downstream (with the current) = \( x + 2.5 \) km/h
- Speed of the boat upstream (against the current) = \( x - 2.5 \) km/h

### Step 4: Calculate the Time Taken for Each Leg of the Journey
- Time taken to travel downstream (to the second ghat) = Distance / Speed
\[ \text{Time downstream} = \frac{21}{x + 2.5} \]
- Time taken to travel upstream (back to the first ghat) = Distance / Speed
\[ \text{Time upstream} = \frac{21}{x - 2.5} \]

### Step 5: Set Up the Equation
The total time spent traveling (downstream + upstream) is equal to the time spent traveling we calculated in Step 2.
\[ \frac{21}{x + 2.5} + \frac{21}{x - 2.5} = 3.83 \]

### Step 6: Solve the Equation
Multiply through by the common denominator \((x + 2.5)(x - 2.5)\) to eliminate the fractions:
\[ 21(x - 2.5) + 21(x + 2.5) = 3.83(x + 2.5)(x - 2.5) \]
This simplifies to:
\[ 21x - 52.5 + 21x + 52.5 = 3.83(x^2 - 6.25) \]
\[ 42x = 3.83x^2 - 23.9375 \]
Rearranging gives:
\[ 3.83x^2 - 42x - 23.9375 = 0 \]

### Step 7: Use the Quadratic Formula
To find \( x \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 3.83, b = -42, c = -23.9375 \).

### Step 8: Calculate the Discriminant and Roots
Calculate the discriminant:
\[ D = (-42)^2 - 4 \cdot 3.83 \cdot (-23.9375) \]
Now calculate \( x \) using the quadratic formula.

### Step 9: Final Calculation
After calculating, we will find the value of \( x \), which is the speed of the boat in still water.