A mini bus takes 6 hours less to cover 1680 km distance if its speed is in creased by 14 km/h?What is the usual time taken by mini bus?

Options

A. 15 h
B. 24 h
C. 25 h
D. 30 h

Answer

To solve the problem step by step, we need to find the usual time taken by the mini bus to cover a distance of 1680 km when its speed is increased by 14 km/h, resulting in a time reduction of 6 hours.

### Step-by-Step Solution:

1. **Define Variables:**
Let the usual speed of the mini bus be \( S \) km/h. Therefore, the usual time taken to cover 1680 km is given by:
\[ \text{Usual Time} = \frac{1680}{S} \]

2. **Increased Speed:**
If the speed is increased by 14 km/h, the new speed becomes:
\[ \text{New Speed} = S + 14 \text{ km/h} \]

3. **New Time Calculation:**
The time taken to cover the same distance at the new speed is:
\[ \text{New Time} = \frac{1680}{S + 14} \]

4. **Set Up the Equation:**
According to the problem, the new time is 6 hours less than the usual time:
\[ \frac{1680}{S} - \frac{1680}{S + 14} = 6 \]

5. **Clear the Fractions:**
To eliminate the fractions, multiply the entire equation by \( S(S + 14) \):
\[ 1680(S + 14) - 1680S = 6S(S + 14) \]

6. **Simplify the Equation:**
Expanding both sides:
\[ 1680S + 23520 - 1680S = 6S^2 + 84S \]
This simplifies to:
\[ 23520 = 6S^2 + 84S \]

7. **Rearranging the Equation:**
Rearranging gives us a standard quadratic equation:
\[ 6S^2 + 84S - 23520 = 0 \]

8. **Divide the Equation:**
To simplify, divide the entire equation by 6:
\[ S^2 + 14S - 3920 = 0 \]

9. **Factor the Quadratic:**
We need to factor or use the quadratic formula to solve for \( S \). The quadratic formula is:
\[ S = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 14, c = -3920 \).

10. **Calculate the Discriminant:**
\[ b^2 - 4ac = 14^2 - 4 \times 1 \times (-3920) = 196 + 15680 = 15876 \]

11. **Find the Roots:**
\[ S = \frac{-14 \pm \sqrt{15876}}{2} \]
Calculate \( \sqrt{15876} \approx 126 \):
\[ S = \frac{-14 \pm 126}{2} \]
This gives two potential solutions:
\[ S = \frac{112}{2} = 56 \quad \text{(valid speed)} \]
\[ S = \frac{-140}{2} = -70 \quad \text{(not valid)} \]

12. **Calculate Usual Time:**
Now that we have \( S = 56 \) km/h, we can find the usual time:
\[ \text{Usual Time} = \frac{1680}{56} = 30 \text{ hours} \]

### Final Answer:
The usual time taken by the mini bus is **30 hours**.