Some friends are leaving a party in Princeton to go home. One group takes a taxi uptown, traveling north at 34 miles per hour. The rest of the friends share another taxi, traveling south at 42 miles per hour. If the two taxis depart at the same time, how much time will pass before they are 10 miles apart?

Question

If necessary, round your answer to the nearest minute.
hours and
minutes

Answer 1

To solve the problem, we need to determine the time it takes for the two groups in taxis to be 10 miles apart.

Let's denote the time passed as \( t \) (in hours).

1. The first taxi moves north at a speed of 34 miles per hour. In \( t \) hours, the distance it travels is: \[ \text{Distance}_{\text{north}} = 34t \]

2. The second taxi moves south at a speed of 42 miles per hour. In \( t \) hours, the distance it travels is: \[ \text{Distance}_{\text{south}} = 42t \]

Since they are traveling in opposite directions, the total distance between the two taxis after time \( t \) is the sum of the distances traveled by both taxis: \[ \text{Total distance} = \text{Distance}{\text{north}} + \text{Distance}{\text{south}} = 34t + 42t = 76t \]

We want this total distance to equal 10 miles: \[ 76t = 10 \]

Now, we can solve for \( t \): \[ t = \frac{10}{76} = \frac{5}{38} \text{ hours} \]

Next, we convert \( \frac{5}{38} \) hours into minutes. Since there are 60 minutes in one hour: \[ \text{Minutes} = \frac{5}{38} \times 60 \approx 7.89 \text{ minutes} \]

Rounding this to the nearest minute, we get approximately 8 minutes.

Thus, the time before the two taxis are 10 miles apart is: \[ \text{Time} = 0 \text{ hours and } 8 \text{ minutes} \]

Answer: 0 hours and 8 minutes