A car is to be driven 200 kms on a highway at an uniform speed of x km/hr (speed limit of the highway
litre per
require 40 <x< 70). The cost of diesel is Rs. 30/litre and is consumed at the rate of 100+
x^2/60 litre per
hour. If the wage of the driver is Rs. 200 per hour then the most economical speed to drive the car
within the speed limit is
Answers
Given : A car is to be driven 200 kms on a highway at an uniform speed of x km/hr (speed limit of the highway require 40 <x< 70 )
The cost of diesel is Rs. 30/litre and is consumed at the rate of 100+
x^2/60 litre per hr
wage of the driver is Rs. 200 per hour
To Find : the most economical speed to drive the car within the speed limit is
Solution:
speed = x km/hr 40 < x < 70
Distance = 200 km
Time = 200/x hr
Diesel = 100 + x²/60 litre per hr
=> Diesel in 200/x hr = ( 100 + x²/60)(200/x)
=> 20000/x + 10x/3 litre
cost of diesel is Rs. 30/litre
Cost of Diesel = 30(20000/x + 10x/3 )
= 600000/x + 100x
Cost of Driver = 200 *(200/x) = 40000/x
Total Cost C = 640000/x + 100x
dC/dx = - 640000/x² + 100
=> - 640000/x² + 100 = 0
=> x² = 6400
=> x = 80
d²C/dx² = 2 . 640000/ x³ > 0
Hence Cost is maximum when x = 80
But need to drive in speed limit 40 < x < 70
Hence most economical speed to drive the car within the speed limit is 70 km/hr
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