A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is –

Options

A. 2mph
B. 2.5 mph
C. 3 mph
D. 4 mph

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define Variables
Let the speed of the stream be \( x \) mph. The speed of the boat in still water is given as 10 mph.

### Step 2: Determine Speeds
- **Downstream Speed**: When the boat is going downstream, the speed is the sum of the boat's speed and the stream's speed:
\[ \text{Downstream Speed} = 10 + x \text{ mph} \]
- **Upstream Speed**: When the boat is going upstream, the speed is the difference between the boat's speed and the stream's speed:
\[ \text{Upstream Speed} = 10 - x \text{ mph} \]

### Step 3: Write the Time Equations
The time taken to travel a distance is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For both downstream and upstream, the distance is 36 miles.

- **Time Downstream**:
\[ T_d = \frac{36}{10 + x} \]
- **Time Upstream**:
\[ T_u = \frac{36}{10 - x} \]

### Step 4: Set Up the Equation
According to the problem, the time taken to travel downstream is 90 minutes less than the time taken to travel upstream. We convert 90 minutes to hours:
\[ 90 \text{ minutes} = \frac{90}{60} = 1.5 \text{ hours} \]
Thus, we can set up the equation:
\[ T_u - T_d = 1.5 \]
Substituting the expressions for \( T_u \) and \( T_d \):
\[ \frac{36}{10 - x} - \frac{36}{10 + x} = 1.5 \]

### Step 5: Solve the Equation
To solve the equation, first, we can multiply through by the common denominator \((10 - x)(10 + x)\):
\[ 36(10 + x) - 36(10 - x) = 1.5(10 - x)(10 + x) \]
This simplifies to:
\[ 36x + 36x = 1.5(100 - x^2) \]
\[ 72x = 150 - 1.5x^2 \]
Rearranging gives us:
\[ 1.5x^2 + 72x - 150 = 0 \]
Dividing through by 1.5 to simplify:
\[ x^2 + 48x - 100 = 0 \]

### Step 6: Solve the Quadratic Equation
Now we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 48, c = -100 \):
\[ x = \frac{-48 \pm \sqrt{48^2 - 4 \cdot 1 \cdot (-100)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{-48 \pm \sqrt{2304 + 400}}{2} \]
\[ x = \frac{-48 \pm \sqrt{2704}}{2} \]
\[ x = \frac{-48 \pm 52}{2} \]
Calculating the two possible values:
1. \( x = \frac{4}{2} = 2 \)
2. \( x = \frac{-100}{2} = -50 \) (not a valid speed)

Thus, the speed of the stream is:
\[ \boxed{2} \text{ mph} \]