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A ranch hand herded his cattle to some grazing land. Then he began riding back to the ranch, which is due west, at $3$ miles per hour. Simultaneously, the cows began moving away from him, too, headed due east at $1$ mile per hour. How long did it take for the herd and the ranch hand to be $3$ miles apart? If necessary, round your answer to the nearest minute. $\newline$ hours and minutes

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Rate of travel: word problems

Full solution

Q. A ranch hand herded his cattle to some grazing land. Then he began riding back to the ranch, which is due west, at

3

miles per hour. Simultaneously, the cows began moving away from him, too, headed due east at

1

mile per hour. How long did it take for the herd and the ranch hand to be

3

miles apart? If necessary, round your answer to the nearest minute.

\newline

____ hours and ____ minutes

Find Relative Speed: We need to find the time it takes for the ranch hand and the cattle to be $3$ miles apart. The ranch hand is moving west at $3$ miles per hour, and the cattle are moving east at $1$ mile per hour. Since they are moving in opposite directions, we can add their speeds to find the relative speed between them.
Calculate Combined Speed: Calculate the combined speed of the ranch hand and the cattle. $\newline$ Combined speed $= \text{Speed of ranch hand} + \text{Speed of cattle}$ $\newline$ Combined speed $= 3 \, \text{mph} + 1 \, \text{mph}$ $\newline$ Combined speed $= 4 \, \text{mph}$
Find Time Taken: Now, we need to find the time it takes for them to be $3$ miles apart at the combined speed of $4$ miles per hour. $\newline$ Time $=$ Distance $/$ Speed $\newline$ Time $=$ $3$ miles $/$ $4$ miles per hour $\newline$ Time $=$ $0.75$ hours
Convert Time to Minutes: To convert the time from hours to minutes, we multiply by $60$ minutes per hour. $\newline$ Time in minutes = Time in hours $\times 60$ minutes per hour $\newline$ Time in minutes = $0.75$ hours $\times 60$ minutes per hour $\newline$ Time in minutes = $45$ minutes
Final Answer: Since $0.75$ hours is exactly $45$ minutes, there is no need to round the answer.

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