A car travelled first 36 km at 6 km/h faster then the usaul speed but it returned the same distance at 6 km/h slower than the usual speed if the total time taken by car is 8 hours for how many hours does it travelled at the faster speed?

Options

A. 4
B. 3
C. 2
D. 1

Answer

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

### Step 1: Define the usual speed
Let the usual speed of the car be \( x \) km/h.

**Hint:** Start by defining the variables that represent the speeds involved in the problem.

### Step 2: Determine the faster and slower speeds
The car travels at a speed of \( x + 6 \) km/h when going and \( x - 6 \) km/h when returning.

**Hint:** Identify the speeds for both parts of the journey based on the information given in the question.

### Step 3: Write the time taken for each part of the journey
The time taken to travel 36 km at the faster speed is:
\[ \text{Time}_{\text{going}} = \frac{36}{x + 6} \]
The time taken to return 36 km at the slower speed is:
\[ \text{Time}_{\text{returning}} = \frac{36}{x - 6} \]

**Hint:** Use the formula for time, which is distance divided by speed, to express the time taken for each leg of the journey.

### Step 4: Set up the equation for total time
According to the problem, the total time taken for the round trip is 8 hours. Therefore, we can write the equation:
\[ \frac{36}{x + 6} + \frac{36}{x - 6} = 8 \]

**Hint:** Combine the times for both journeys into a single equation to represent the total time.

### Step 5: Simplify the equation
Multiply through by the common denominator \((x + 6)(x - 6)\) to eliminate the fractions:
\[ 36(x - 6) + 36(x + 6) = 8(x^2 - 36) \]
This simplifies to:
\[ 36x - 216 + 36x + 216 = 8x^2 - 288 \]
Combining like terms gives:
\[ 72x = 8x^2 - 288 \]
Rearranging leads to:
\[ 8x^2 - 72x - 288 = 0 \]

**Hint:** When simplifying, ensure you combine like terms correctly and rearrange the equation to standard quadratic form.

### Step 6: Divide the entire equation by 8
To simplify further:
\[ x^2 - 9x - 36 = 0 \]

**Hint:** Dividing by a common factor can make calculations easier.

### Step 7: Factor the quadratic equation
Now we can factor the quadratic:
\[ (x - 12)(x + 3) = 0 \]
This gives us the solutions:
\[ x = 12 \quad \text{or} \quad x = -3 \]

**Hint:** Factor the quadratic to find the possible values for the usual speed.

### Step 8: Determine the valid speed
Since speed cannot be negative, we take \( x = 12 \) km/h as the usual speed.

**Hint:** Always consider the context of the problem when selecting values from your solutions.

### Step 9: Calculate the faster speed
The faster speed is:
\[ x + 6 = 12 + 6 = 18 \text{ km/h} \]

**Hint:** Use the value of the usual speed to find the other speeds mentioned in the problem.

### Step 10: Calculate the time taken at the faster speed
The time taken to travel 36 km at the faster speed is:
\[ \text{Time}_{\text{faster}} = \frac{36}{18} = 2 \text{ hours} \]

**Hint:** Use the distance and the faster speed to find the time taken for that part of the journey.

### Final Answer
The car traveled for **2 hours** at the faster speed.

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