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