The current of a stream runs at 1 km/hr. A motorboat goes 35 km upstream and back again to the starting point in 12 hours. The speed of motor boat in still water is -----

Options

A. 6 km/hr
B. 7 km/hr
C. 8.5 km/hr
D. 8 km/hr

Answer

To find the speed of the motorboat in still water, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = speed of the motorboat in still water (in km/hr)
- The speed of the current \( y = 1 \) km/hr

### Step 2: Determine Speeds
The speed of the motorboat downstream (with the current) is:
\[ x + y = x + 1 \text{ km/hr} \]
The speed of the motorboat upstream (against the current) is:
\[ x - y = x - 1 \text{ km/hr} \]

### Step 3: Set Up the Time Equation
The total distance traveled is 35 km upstream and 35 km downstream. The total time taken for the round trip is 12 hours. We can express the time taken for each leg of the journey as:
- Time taken to go upstream:
\[ \text{Time upstream} = \frac{35}{x - 1} \]
- Time taken to go downstream:
\[ \text{Time downstream} = \frac{35}{x + 1} \]

### Step 4: Write the Total Time Equation
The total time for the trip is the sum of the time taken upstream and downstream:
\[ \frac{35}{x - 1} + \frac{35}{x + 1} = 12 \]

### Step 5: Simplify the Equation
Multiply through by \( (x - 1)(x + 1) \) to eliminate the denominators:
\[ 35(x + 1) + 35(x - 1) = 12(x^2 - 1) \]
This simplifies to:
\[ 35x + 35 + 35x - 35 = 12x^2 - 12 \]
\[ 70x = 12x^2 - 12 \]

### Step 6: Rearrange to Form a Quadratic Equation
Rearranging gives us:
\[ 12x^2 - 70x - 12 = 0 \]

### Step 7: Solve the Quadratic Equation
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 12, b = -70, c = -12 \):
1. Calculate the discriminant:
\[ b^2 - 4ac = (-70)^2 - 4 \cdot 12 \cdot (-12) = 4900 + 576 = 5476 \]
2. Calculate the roots:
\[ x = \frac{70 \pm \sqrt{5476}}{24} \]
\[ \sqrt{5476} = 74 \]
Thus,
\[ x = \frac{70 + 74}{24} \quad \text{or} \quad x = \frac{70 - 74}{24} \]
\[ x = \frac{144}{24} = 6 \quad \text{or} \quad x = \frac{-4}{24} = -\frac{1}{6} \text{ (not valid)} \]

### Step 8: Conclusion
The speed of the motorboat in still water is:
\[ \boxed{6 \text{ km/hr}} \]