To solve the problem of determining when the second driver will overtake the first driver, we need to compare the distances each driver has traveled at any point in time. Here’s a step-by-step approach:
1. Assign Variables for Speeds and Time Difference:
- Speed of the first driver: [tex]\( 52 \)[/tex] miles per hour
- Speed of the second driver: [tex]\( 68 \)[/tex] miles per hour
- Time difference between their departures: [tex]\( 2 \)[/tex] hours
2. Set up the Distance Equations:
- Let [tex]\( t \)[/tex] be the number of hours the second driver has been driving when they overtake the first driver.
- By the time the second driver has been driving for [tex]\( t \)[/tex] hours, the first driver has been driving for [tex]\( t + 2 \)[/tex] hours (since the first driver left 2 hours earlier).
3. Formulate Distance Traveled Equations:
- The distance traveled by the first driver after [tex]\( t + 2 \)[/tex] hours is: [tex]\( \text{Distance}_1 = 52 \times (t + 2) \)[/tex]
- The distance traveled by the second driver after [tex]\( t \)[/tex] hours is: [tex]\( \text{Distance}_2 = 68 \times t \)[/tex]
4. Set the Distances Equal to Find the Overtaking Point:
- When the second driver overtakes the first driver, both will have traveled the same distance.
- Therefore, [tex]\( 52(t + 2) = 68t \)[/tex]
5. Solve the Equation:
- Expand the left-hand side: [tex]\( 52t + 104 = 68t \)[/tex]
- Move all [tex]\( t \)[/tex]-terms to one side by subtracting [tex]\( 52t \)[/tex] from both sides: [tex]\( 104 = 68t - 52t \)[/tex]
- Simplify: [tex]\( 104 = 16t \)[/tex]
- Solve for [tex]\( t \)[/tex] by dividing both sides by [tex]\( 16 \)[/tex]: [tex]\( t = \frac{104}{16} \)[/tex]
- Simplify the fraction: [tex]\( t = 6.5 \)[/tex]
Thus, it will take the second driver [tex]\( 6.5 \)[/tex] hours to overtake the first driver after the second driver starts traveling.
