Question

two hours after harlan had left his house for his out of state business trip, his wife maggie noticed that he had forgotten his briefcase. Knowing that the briefcase contained all of his important papers, she immediately hopped into her car and began to chase after him at a speed of 65 mph. If harlan drove 55 mph the entire time, how many hours did it take maggie to catch him?

Answer 1

Let's break this problem down step by step.

1. Determine Harlan's head start: Harlan left two hours earlier than Maggie. In those two hours, he drove at a speed of 55 mph. Therefore, the distance he covered during that time is: \[ \text{Distance} = \text{Speed} \times \text{Time} = 55 , \text{mph} \times 2 , \text{hours} = 110 , \text{miles} \]

2. Set up the equations: Let \( t \) be the time (in hours) it takes for Maggie to catch up to Harlan after she starts driving. During this time, Maggie drives at 65 mph and Harlan continues to drive at 55 mph.

   The distance Maggie covers in \( t \) hours is: \[ \text{Distance}_{\text{Maggie}} = 65t \]

   The distance Harlan covers during the same time \( t \) (after Maggie starts) plus his head start of 110 miles is: \[ \text{Distance}_{\text{Harlan}} = 110 + 55t \]

3. Set the distances equal: Since Maggie catches up to Harlan when the distances are equal, we set the two distances equal to each other: \[ 65t = 110 + 55t \]

4. Solve for \( t \): Rearranging the equation to isolate \( t \): \[ 65t - 55t = 110 \] \[ 10t = 110 \] \[ t = \frac{110}{10} = 11 , \text{hours} \]

Therefore, it took Maggie 11 hours to catch up to Harlan after she started driving. The total time from when Harlan left until Maggie caught up with him is \( 2 + 11 = 13 \) hours.