A bus starts running with the initial speed of 21 km/hr and its speed increases evry hour by 3 km/kr . How many hours will it take to cover a distance of 252 km ?

Options

A. 3
B. 5
C. 8
D. 10

Answer

To solve the problem step by step, we need to determine how long it will take for a bus to cover a distance of 252 km, given that it starts with an initial speed of 21 km/hr and increases its speed by 3 km/hr every hour.

### Step-by-Step Solution:

1. **Identify the initial speed and acceleration**:
- The initial speed \( u = 21 \) km/hr.
- The speed increases by \( a = 3 \) km/hr every hour.

2. **Determine the speed after each hour**:
- After 1 hour: Speed = \( 21 + 3 \times 1 = 24 \) km/hr.
- After 2 hours: Speed = \( 21 + 3 \times 2 = 27 \) km/hr.
- After 3 hours: Speed = \( 21 + 3 \times 3 = 30 \) km/hr.
- After \( n \) hours: Speed = \( 21 + 3n \) km/hr.

3. **Calculate the distance covered in \( n \) hours**:
- The distance covered in each hour can be expressed as:
- Distance in 1st hour = \( 21 \) km
- Distance in 2nd hour = \( 24 \) km
- Distance in 3rd hour = \( 27 \) km
- Distance in \( n \) hours = \( 21 + 24 + 27 + ... + (21 + 3(n-1)) \)

4. **Use the formula for the sum of an arithmetic series**:
- The total distance \( S \) covered in \( n \) hours can be calculated using the formula for the sum of the first \( n \) terms of an arithmetic series:
\[ S = \frac{n}{2} \times (2a + (n-1)d) \]
where:
- \( a = 21 \) (the first term),
- \( d = 3 \) (the common difference),
- \( S = 252 \) km (the total distance).

5. **Set up the equation**:
\[ 252 = \frac{n}{2} \times (2 \times 21 + (n-1) \times 3) \]
Simplifying this:
\[ 252 = \frac{n}{2} \times (42 + 3n - 3) \]
\[ 252 = \frac{n}{2} \times (39 + 3n) \]
\[ 504 = n(39 + 3n) \]
\[ 3n^2 + 39n - 504 = 0 \]

6. **Simplify the quadratic equation**:
- Divide the entire equation by 3:
\[ n^2 + 13n - 168 = 0 \]

7. **Factor the quadratic equation**:
- We need two numbers that multiply to \(-168\) and add to \(13\). The numbers are \(21\) and \(-8\).
\[ (n + 21)(n - 8) = 0 \]

8. **Solve for \( n \)**:
- Setting each factor to zero gives:
\[ n + 21 = 0 \quad \text{or} \quad n - 8 = 0 \]
- Thus, \( n = -21 \) (not valid) or \( n = 8 \).

9. **Conclusion**:
- The bus will take **8 hours** to cover a distance of 252 km.