Question 322397
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Everything in this problem is based on the <i>distance equals rate times time</i> formula.


Let *[tex \Large d_l] represent the distance traveled on land.


Let *[tex \Large d_w] represent the distance traveled on the water.


Let *[tex \Large t] represent the elapsed time traveling on land.


We know the rate of speed on land because that is given as 60 mph.


The rate of speed for the outbound trip on water is the rate in still water MINUS the rate of the current (because he was traveling AGAINST the current), so 20 mph minus 4 mph is 16 mph.


Likewise, the rate of speed for the return trip on water is 24 mph.


If the total outbound trip took 4.5 hours and the land part of the trip took *[tex \Large t] hours, then the water part of the outbound trip must have taken *[tex \Large 4.5\ -\ t] hours.


Similarly, the water part of the return trip must have taken *[tex \Large 3.5\ -\ t] hours.


Let's put some of this together.  We can describe the outbound trip over water by the relationship:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d_w\ =\ 16(4.5\ -\ t)]


And we can describe the return trip over water as:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d_w\ =\ 24(3.5\ -\ t)]


Having two things both equal to *[tex \Large d_w], we can set them equal to each other:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 16(4.5\ -\ t)\ =\ 24(3.5\ -\ t)]


From which we can solve for *[tex \Large t\ =\ 1.5] hours, the travel time on land.  (Verification of this result is left as an exercise for the student).


Then, knowing that *[tex \Large d_l\ =\ 60t], we can say that *[tex \Large d_l\ = 90]


Further, *[tex \Large t\ =\ 1.5] says that the travel time for the outbound trip on water is *[tex \Large 4.5\ -\ 1.5\ =\ 3], hence the distance over water for the outbound trip is *[tex \Large d_w\ =\ 16\,\cdot\,3\ =\ 48] miles.


As a check, the distance over water for the return trip better be the same as for the outbound trip:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d_w\ =\ 24(3.5\ -\ 1.5)\ =\ 48]


And our answer checks.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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