Q31 of 35 Page 55

From point 'A' located on a highway one has to get by a car as soon as possible to point 'B' located in the field at a distance ' l ' from point 'D'. If the car moves n times slower in the field at what distance x from D one must turn off the highway.


C is the point at which car turns


CD = x and DB = l (Perpendicular distance of B from D)


Let the speed of the car on the field = v


Then speed of car on highway = v × n


Now, car travels on the highway for the distance of d-x and then car moves in the field. Distance moved by the car in the field is. (Using Pythagoras Theorem)


Highway



Distance = d – x


Speed = v × n


Now,


So,


Field



Distance =


Speed = v


Now,


So,


Now, total time taken = T =


T =


Now, for minimizing the time, we differentiate T with respect to x and equate it to zero.








Now, for minima,


Hence at a distance from point D, the car must turn off the highway.


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