To determine when Truck B catches up with Truck A and how far they are from the gas station at that moment, we can break down the problem step by step. Let’s start by analyzing the situation with some basic calculations.
Understanding the Situation
Truck A is traveling at a speed of 50 miles per hour (mph) and passes the gas station at 12:00 PM. Truck B, on the other hand, is moving faster at 70 mph and passes the same gas station at 12:30 PM. This means Truck A has a 30-minute head start before Truck B begins its journey.
Calculating the Head Start
In the 30 minutes (or 0.5 hours) that Truck A is on the road before Truck B starts, we can calculate how far Truck A travels:
- Distance = Speed × Time
- Distance = 50 mph × 0.5 hours = 25 miles
So, by the time Truck B starts, Truck A is already 25 miles ahead of the gas station.
Setting Up the Catch-Up Equation
Now, we need to find out when Truck B will catch up to Truck A. Let’s denote the time it takes for Truck B to catch up to Truck A after 12:30 PM as t hours. During this time, both trucks will be traveling, and we can set up the following equations based on their speeds:
- Distance traveled by Truck A = 50t + 25 (since it already had a 25-mile head start)
- Distance traveled by Truck B = 70t
Finding the Catch-Up Time
To find when Truck B catches up, we set the distances equal to each other:
50t + 25 = 70t
Now, let’s solve for t:
- 25 = 70t - 50t
- 25 = 20t
- t = 25 / 20
- t = 1.25 hours
Calculating the Catch-Up Time
Since Truck B starts at 12:30 PM, we add 1.25 hours to this time:
- 1 hour and 15 minutes after 12:30 PM is 1:45 PM.
Thus, Truck B catches up with Truck A at 1:45 PM.
Determining the Distance from the Gas Station
Next, we need to find out how far both trucks are from the gas station at that time. We can use either truck’s distance formula, but let’s use Truck B’s since it’s simpler:
- Distance = Speed × Time
- Distance = 70 mph × 1.25 hours = 87.5 miles
Therefore, when Truck B catches up with Truck A at 1:45 PM, both trucks are 87.5 miles away from the gas station.
Summary
In summary, Truck B catches up with Truck A at 1:45 PM, and at that moment, both trucks are 87.5 miles from the gas station. This problem illustrates how relative speeds and time can be used to solve real-world scenarios involving motion.
