The speed of a stream is 9 km/hr. A boat goes 21 km and comes back to the starting point in 8 hours. What is the speed (in km/hr) of the boat in still water?

Options

A. 10
B. 12
C. 8
D. 15

Answer

To find the speed of the boat in still water, we can use the information provided in the question. Let's break down the solution step by step.

### Step 1: Understand the Problem
The speed of the stream is given as 9 km/hr. The boat travels a distance of 21 km downstream and then returns upstream, taking a total of 8 hours for the round trip.

### Step 2: Define Variables
Let the speed of the boat in still water be \( b \) km/hr.
- The effective speed of the boat downstream (with the current) is \( b + 9 \) km/hr.
- The effective speed of the boat upstream (against the current) is \( b - 9 \) km/hr.

### Step 3: Calculate Time for Each Leg of the Journey
The time taken to travel downstream (21 km) can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Thus, the time taken to go downstream is:
\[ \text{Time}_{\text{downstream}} = \frac{21}{b + 9} \]

Similarly, the time taken to return upstream is:
\[ \text{Time}_{\text{upstream}} = \frac{21}{b - 9} \]

### Step 4: Set Up the Equation
The total time for the round trip is given as 8 hours. Therefore, we can set up the equation:
\[ \frac{21}{b + 9} + \frac{21}{b - 9} = 8 \]

### Step 5: Solve the Equation
To solve the equation, we first find a common denominator:
\[ (b + 9)(b - 9) \]
Thus, we rewrite the equation:
\[ 21(b - 9) + 21(b + 9) = 8(b^2 - 81) \]
Simplifying this, we get:
\[ 21b - 189 + 21b + 189 = 8b^2 - 648 \]
Combining like terms:
\[ 42b = 8b^2 - 648 \]
Rearranging gives us:
\[ 8b^2 - 42b - 648 = 0 \]

### Step 6: Simplify the Quadratic Equation
Dividing the entire equation by 2 to simplify:
\[ 4b^2 - 21b - 324 = 0 \]

### Step 7: Use the Quadratic Formula
Using the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):
Here, \( A = 4 \), \( B = -21 \), and \( C = -324 \).
Calculating the discriminant:
\[ B^2 - 4AC = (-21)^2 - 4 \times 4 \times (-324) = 441 + 5184 = 5625 \]
Taking the square root:
\[ \sqrt{5625} = 75 \]
Now substituting back into the formula:
\[ b = \frac{21 \pm 75}{8} \]
Calculating the two possible values:
1. \( b = \frac{96}{8} = 12 \)
2. \( b = \frac{-54}{8} \) (not valid since speed cannot be negative)

### Step 8: Conclusion
Thus, the speed of the boat in still water is:
\[ \boxed{12 \text{ km/hr}} \]