Question 1188745
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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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<pre>
(1)  <U>kinematic technique</U>


     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/120}}}  hours.

     The time for the second half of the trip (returning back) is  {{{d/80}}}  hours.

     Total time is the sum  {{{d/120}}} + {{{d/80}}}.

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


        average speed = {{{(2d)/(d/120 + d/80)}}} = canceling d in both numerator and denominator = {{{2/(1/120+1/80)}}} = 

                      = {{{(2*120*80)/(80+120)}}} = {{{(2*120*80)/200}}} = {{{(2*12*8)/2}}} = 12*8 = 96 km/h.    <U>ANSWER</U>



     <U>harmonic mean technique</U>


     Use the harmonic mean formula for the average speed

        average speed = {{{2*v[1]*v[2]/(v[1]+v[2])}}} = {{{2*120*80)/(80+120)}}} = 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[1]}}}  and {{{v[2]}}} for each part.




(2)  tap P fills {{{1/6}}} of the tank volume per hour.

     tap Q fills {{{1/9}}} of the tank volume per hour.

     Working together, both taps fill  {{{1/6}}} + {{{1/9}}} = {{{6/36 + 4/36}}} = {{{10/36}}} = {{{5/18}}}  of the tank volume.

     Hence, the tank will be filled in  {{{18/5}}} = 3 {{{3/5}}}  hours = 3 hours and 36 minutes, if both taps work simulaneously.
</pre>

Solved.