The speed of a stream is 3 km/hr. A boat goes 36 km and comes back to the starting point in 9 hours. What is the speed (in km/hr) of the boat in still water?

Options

A. 8
B. 9
C. 10
D. 7

Answer

To solve the problem step by step, we need to find the speed of the boat in still water. Let's denote the speed of the boat in still water as \( x \) km/hr.

### Step 1: Understand the scenario
The speed of the stream is given as 3 km/hr. When the boat is going downstream (with the current), its effective speed will be \( x + 3 \) km/hr. When the boat is going upstream (against the current), its effective speed will be \( x - 3 \) km/hr.

### Step 2: Calculate time taken for each journey
The distance traveled by the boat in each direction is 36 km. The total time taken for the round trip is given as 9 hours.

- Time taken to go downstream:
\[ \text{Time}_{\text{downstream}} = \frac{36}{x + 3} \]

- Time taken to come back upstream:
\[ \text{Time}_{\text{upstream}} = \frac{36}{x - 3} \]

### Step 3: Set up the equation
The total time for the round trip is the sum of the time taken to go downstream and the time taken to come back upstream:
\[ \frac{36}{x + 3} + \frac{36}{x - 3} = 9 \]

### Step 4: Solve the equation
To solve this equation, we can first multiply through by the common denominator \((x + 3)(x - 3)\) to eliminate the fractions:
\[ 36(x - 3) + 36(x + 3) = 9(x + 3)(x - 3) \]

Expanding both sides:
\[ 36x - 108 + 36x + 108 = 9(x^2 - 9) \]
\[ 72x = 9x^2 - 81 \]

Rearranging gives us:
\[ 9x^2 - 72x - 81 = 0 \]

### Step 5: Simplify the quadratic equation
Dividing the entire equation by 9:
\[ x^2 - 8x - 9 = 0 \]

### Step 6: Factor the quadratic equation
Now we can factor the quadratic:
\[ (x - 9)(x + 1) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \quad (\text{not valid since speed cannot be negative}) \]

### Conclusion
Thus, the speed of the boat in still water is:
\[ \boxed{9} \text{ km/hr} \]