A bus starts running with the initial speed of 6 km/hr and its speed increases every hour by certain amount. If it takes 14 hours to cover a distance of 630 km, then what will be hourly increment in the speed of bus?

Correct option is D

Given:
Initial speed of the bus = 6 km/hr
Time taken to cover the distance = 14 hours
Total distance covered = 630 km
The speed of the bus increases by a certain amount every hour.
Formula Used:
The total distance covered can be calculated using the sum of the arithmetic series:
​$$S_n = \frac{n}{2} \left( 2a + (n - 1) d \right)$$​​
Where:
$$S_n$$​ is the total distance (630 km)
n is the total time (14 hours)
a is the initial speed (6 km/hr)
d is the common difference (the hourly increment x)
Solution:

Let the hourly increment in speed be x km/hr.
Applying the formula for the sum of the first n terms of an arithmetic progression:
​$$S_n = \frac{n}{2} \left( 2a + (n-1) \times d \right)$$​​
​$$630 = \frac{14}{2} \left( 2 \times 6 + (14 - 1) \times x \right)$$​​
​$$630 = 7 \left( 12 + 13x \right)$$​​
630 = 84 + 91x
630 - 84 = 91x
546 = 91x
x = $$\frac{546}{91}$$​ = 6 km/hr
Thus, the correct option is (d) 6 km/hr