SOLUTION: 1:A car traveled from Dar es salaam to morogoror at 120km/h and returned to dar es salaam at 80km/h.find average speed for the entire journey.(use both kinematic and harmonic mean

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Question 1188745: 1:A car traveled from Dar es salaam to morogoror at 120km/h and returned to dar es salaam at 80km/h.find average speed for the entire journey.(use both kinematic and harmonic mean technique)
2:tap P can fill a tank in 6 hours and tap Q can fill the same tank in 9 hours.if the two taps are turned at the same time, how long will the two taps take to fill the tank
Answer by ikleyn(52778) About Me  (Show Source):
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(1) A car traveled from Dar es salaam to morogoror at 120km/h and returned to dar es salaam at 80km/h.
find average speed for the entire journey.(use both kinematic and harmonic mean technique)
(2) tap P can fill a tank in 6 hours and tap Q can fill the same tank in 9 hours.
if the two taps are turned at the same time, how long will the two taps take to fill the tank
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(1)  kinematic technique


     Let d be one way distance (in kilometers).

     Then the entire trip is 2d kilometers.


     The time for the first half of the trip is  d%2F120  hours.

     The time for the second half of the trip (returning back) is  d%2F80  hours.

     Total time is the sum  d%2F120 + d%2F80.

     Average soeed for the entire trip is the total distance 2d divided by the total time


        average speed = %282d%29%2F%28d%2F120+%2B+d%2F80%29 = canceling d in both numerator and denominator = 2%2F%281%2F120%2B1%2F80%29 = 

                      = %282%2A120%2A80%29%2F%2880%2B120%29 = %282%2A120%2A80%29%2F200 = %282%2A12%2A8%29%2F2 = 12*8 = 96 km/h.    ANSWER



     harmonic mean technique


     Use the harmonic mean formula for the average speed

        average speed = 2%2Av%5B1%5D%2Av%5B2%5D%2F%28v%5B1%5D%2Bv%5B2%5D%29 = 2%2A120%2A80%29%2F%2880%2B120%29 = 96 km/h


     which we, actually, derived in the kinematic solution.


     This harmonic mean formula is valid for any trip, consisting of two equal parts at given average velocities 
     v%5B1%5D  and v%5B2%5D for each part.




(2)  tap P fills 1%2F6 of the tank volume per hour.

     tap Q fills 1%2F9 of the tank volume per hour.

     Working together, both taps fill  1%2F6 + 1%2F9 = 6%2F36+%2B+4%2F36 = 10%2F36 = 5%2F18  of the tank volume.

     Hence, the tank will be filled in  18%2F5 = 3 3%2F5  hours = 3 hours and 36 minutes, if both taps work simulaneously.

Solved.