To solve this problem, we need to determine how long it takes for the car to cover a distance of 345 km by gradually increasing its speed every 2 hours.

Let’s break down the problem into intervals and calculate the time taken for each interval.

Interval 1:

Initial speed = 70 km/hr

Time taken = 2 hours

Distance covered = Speed × Time = 70 km/hr × 2 hr = 140 km

Interval 2:

Speed after 2 hours = 70 km/hr + 10 km/hr = 80 km/hr

Time taken = 2 hours

Distance covered = Speed × Time = 80 km/hr × 2 hr = 160 km

Interval 3:

Speed after 4 hours = 80 km/hr + 10 km/hr = 90 km/hr

Time taken = 2 hours

Distance covered = Speed × Time = 90 km/hr × 2 hr = 180 km

Total distance covered in the above intervals = 140 km + 160 km + 180 km = 480 km

Now, we need to determine how much time is needed to cover the remaining distance of 345 km (480 km – 345 km = 135 km) with a constant speed of 90 km/hr.

Time taken for the remaining distance = Distance / Speed = 135 km / 90 km/hr = 1.5 hours

Adding the time taken for the intervals and the remaining distance, we get:

Total time taken = 2 hours + 2 hours + 2 hours + 1.5 hours = 7.5 hours

Converting 0.5 hours into minutes, we have 0.5 hours × 60 minutes/hour = 30 minutes.

Therefore, the car will cover 345 km in 7 hours and 30 minutes.

The closest option to this answer is (b) 4 hr 5 min.