The distance an automobile can travel in h hours at an average rate of 50 miles an hour is found y the formula d=50h. Problem: A car was driven for 3 hours. The driver then stopped 3/4 of an hour for lunch. After lunch he drove for 2 and 1/2 hours, then stopped 1/2 hr for a flat tire. After that he resumed driving the car. There is another car that starts from the same place 3 hrs after the first car and overtakes the first in 5hrs. Determine the rate of the second car. How far apart will the cars be 1 and 1/2 hours after the second car starts?

To find the rate of the second car, we need to determine the distance traveled by both cars during the given time frame.

For the first car:
Total driving time = 3 + 2.5 = 5.5 hours
Total stops = 3/4 + 1/2 = 1.25 hours

The first car's driving time, excluding stops, is 5.5 - 1.25 = 4.25 hours.

Using the formula d = 50h, we can calculate the distance traveled by the first car:
d1 = 50 * 4.25 = 212.5 miles

Now, let's determine the time it takes the second car to overtake the first car.

Using the same formula, d = 50h, the second car overtakes the first car in 5 hours.
So, the distance traveled by the second car is also 5 * 50 = 250 miles.

Now, let's calculate how far apart the cars will be 1.5 hours after the second car starts.

The first car has already been traveling for 3 + 5 = 8 hours.
At a constant rate of 50 mph, the first car's distance after 8 hours is 50 * 8 = 400 miles.

The second car drives for an additional 1.5 hours at an unknown rate, which we'll call r.
Therefore, the distance covered by the second car in this time is 1.5r.

The total distance between the cars after 1.5 hours is given by:
400 - (212.5 + 1.5r)

To determine r, we can set this expression equal to the distance traveled by the second car:
400 - (212.5 + 1.5r) = 250

Now, let's solve for r:

400 - 212.5 - 1.5r = 250
187.5 - 1.5r = 250
-1.5r = 250 - 187.5
-1.5r = 62.5
r = 62.5 / -1.5
r = -41.67

Since a negative rate is not meaningful in this context, we can ignore this value and conclude that the rate of the second car cannot be determined with the given information.

To determine the rate of the second car, we need to first calculate the distance traveled by the first car. We can use the given formula d=50h, where d represents the distance traveled and h represents the time in hours.

First, let's calculate the distance traveled by the first car in each segment:

Segment 1: The first car drives for 3 hours, so the distance traveled is d1 = 50 * 3 = 150 miles.
Lunch Break: The first car stops for 3/4 hour, so there is no distance traveled during this time.
Segment 2: The first car drives for 2 1/2 hours, so the distance traveled is d2 = 50 * (2 + 1/2) = 125 miles.
Flat Tire Break: The first car stops for 1/2 hour, so there is no distance traveled during this time.

The total distance traveled by the first car can be calculated as follows:
Total distance = d1 + d2 = 150 + 125 = 275 miles.

Now, let's determine the rate of the second car. We know that the second car overtakes the first car in 5 hours. The distance traveled by the second car is the same as the distance traveled by the first car, which is 275 miles.

Rate of the second car = Total distance traveled / Time taken = 275 miles / 5 hours = 55 miles per hour.

To find the distance between the two cars 1 and 1/2 hours after the second car starts, we need to calculate the distance traveled by each car during this time.

The first car has been driving for a total of 3 + 3/4 + 2 1/2 + 1/2 = 6 3/4 hours.
Therefore, the distance traveled by the first car is d1 = 50 * (6 + 3/4) = 337.5 miles.

The second car has been driving for 1 1/2 hours.
Therefore, the distance traveled by the second car is d2 = 55 * (1 + 1/2) = 82.5 miles.

The distance between the two cars after 1 1/2 hours is given by:
Distance between the cars = Distance traveled by the first car - Distance traveled by the second car
= 337.5 - 82.5
= 255 miles.

Therefore, the cars will be 255 miles apart 1 and 1/2 hours after the second car starts.