A man commutes to work a distance of 40 miles and returns on the same route at the end of the day. His average rate on the retum trip is 15 miles per hour faster than his average rate on the outgoing trip. Write the total time, T, in hours, devoted to his outgoing and retum trips as a function of his rate on the outgoing trip, \( x \). Then find and interpret \( \mathrm{T}(30) \). \[ \text { Hint: Time traveled }=\frac{\text { Distance traveled }}{\text { Rate of travel }} \] \[ I(30)=\overline{9} \text { (Simplity your answer.) } \] What does \( T(30) \) mean? A. It is the total time devoted to the man's commute when his rate of travel for the outgoing trip is 30 miles per hour. B. It is the total time devoted to the man's commute when his rate of travel for the return trip is 30 miles per hour faster then his rate of travel for the outgoing trip. C. It is the total time devoted to the man's commute when his rate of travel for the outgoing trip is 30 miles per hour faster then his rate of travel for the return trip. D. It is the average rate of travel for the two trips combined.
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To find the total time \( T \) devoted to the man's commute, we can break it down into two parts: the outgoing trip and the return trip. If the outgoing trip rate is \( x \) miles per hour, then the time for the outgoing trip is \( \frac{40}{x} \) hours. The return trip rate is \( x + 15 \) miles per hour, which means the time for the return trip is \( \frac{40}{x + 15} \) hours. Therefore, the total time \( T \) as a function of \( x \) is: \[ T(x) = \frac{40}{x} + \frac{40}{x + 15}. \] Now, calculating \( T(30) \): \[ T(30) = \frac{40}{30} + \frac{40}{30 + 15} = \frac{4}{3} + \frac{40}{45} = \frac{4}{3} + \frac{8}{9} = \frac{12}{9} + \frac{8}{9} = \frac{20}{9} \text{ hours (or approximately } 2.22 \text{ hours).} \] This means \( T(30) \) indicates the total time spent commuting when the rate for the outgoing trip is 30 miles per hour. A. It is the total time devoted to the man's commute when his rate of travel for the outgoing trip is 30 miles per hour.
