Let's define the problem:
- The ranch hand is riding back to the ranch at a speed of 3 miles per hour (mph) toward the west.
- The cows are moving away from the ranch hand at a speed of 1 mph toward the east.
Since their movements are in opposite directions, we can add their speeds to find the rate at which the distance between them increases. The combined speed at which the distance between the ranch hand and the herd increases is:
\[ 3 \text{ mph} + 1 \text{ mph} = 4 \text{ mph} \]
We want to find the time it takes for the distance between them to reach 3 miles. We can use the formula for distance, which is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Let \( t \) be the time in hours. We can set up the equation for the distance:
\[ 4 \text{ mph} \times t = 3 \text{ miles} \]
Now, we solve for \( t \):
\[ t = \frac{3 \text{ miles}}{4 \text{ mph}} = \frac{3}{4} \text{ hours} \]
Next, we convert \( \frac{3}{4} \) hours into minutes:
\[ \frac{3}{4} \text{ hours} = \frac{3}{4} \times 60 \text{ minutes} = 45 \text{ minutes} \]
Thus, the time it takes for the herd and the ranch hand to be 3 miles apart is:
0 hours and 45 minutes.