Time taken by a boat to cover 162 km each in downstream and in upstream is 14 hours and 24 minutes. If speed of stream is 6 km/hr., then find the time taken by boat to cover 240 km in upstream.

Options

A. A)`7(1)/(3)"hours"`
B. B)`18(2)/(3)"hours"`
C. C)`13(1)/(3)"hours"`
D. D)`16(2)/(3)"hours"`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Convert Total Time to Hours
The total time taken by the boat to cover 162 km downstream and 162 km upstream is given as 14 hours and 24 minutes. First, we need to convert this time into hours.

**Calculation:**
\[ 14 \text{ hours} + \frac{24 \text{ minutes}}{60} = 14 + \frac{2}{5} = \frac{72}{5} \text{ hours} \]

**Hint:** Remember to convert minutes into hours by dividing by 60.

### Step 2: Define Variables
Let:
- \( B \) = speed of the boat in still water (in km/hr)
- Speed of the stream = 6 km/hr

### Step 3: Write Down the Time Equations
The time taken to travel downstream and upstream can be expressed as:
- Downstream time = \( \frac{162}{B + 6} \)
- Upstream time = \( \frac{162}{B - 6} \)

According to the problem, the total time for both journeys is:
\[ \frac{162}{B + 6} + \frac{162}{B - 6} = \frac{72}{5} \]

**Hint:** Use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to set up your equations.

### Step 4: Simplify the Equation
Multiply through by the common denominator \( (B + 6)(B - 6) \):
\[ 162(B - 6) + 162(B + 6) = \frac{72}{5} (B^2 - 36) \]

This simplifies to:
\[ 162B - 972 + 162B + 972 = \frac{72}{5} (B^2 - 36) \]
\[ 324B = \frac{72}{5} (B^2 - 36) \]

**Hint:** When multiplying, ensure you distribute correctly and combine like terms.

### Step 5: Rearrange to Form a Quadratic Equation
Rearranging gives:
\[ 72B^2 - 1620B - 2592 = 0 \]

Dividing the entire equation by 36 simplifies it to:
\[ 2B^2 - 45B - 72 = 0 \]

**Hint:** Always check if you can simplify the equation to make calculations easier.

### Step 6: Factor the Quadratic Equation
To factor the quadratic equation \( 2B^2 - 45B - 72 = 0 \):
\[ (2B + 3)(B - 24) = 0 \]

Setting each factor to zero gives:
1. \( 2B + 3 = 0 \) → \( B = -\frac{3}{2} \) (ignore this as speed cannot be negative)
2. \( B - 24 = 0 \) → \( B = 24 \) km/hr

**Hint:** When factoring, look for pairs of numbers that multiply to give the constant term and add to give the coefficient of \( B \).

### Step 7: Calculate Upstream Speed
Now, calculate the upstream speed:
\[ \text{Upstream speed} = B - \text{speed of stream} = 24 - 6 = 18 \text{ km/hr} \]

**Hint:** Remember to subtract the speed of the stream from the speed of the boat to find the effective speed upstream.

### Step 8: Calculate Time to Cover 240 km Upstream
Finally, calculate the time taken to cover 240 km upstream:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{240}{18} = \frac{40}{3} \text{ hours} \]

Converting \( \frac{40}{3} \) hours into hours and minutes:
\[ \frac{40}{3} = 13 \frac{1}{3} \text{ hours} \]

**Hint:** To convert an improper fraction to a mixed number, divide the numerator by the denominator.

### Final Answer
The time taken by the boat to cover 240 km upstream is **13 hours and 20 minutes** or **13 1/3 hours**.