Answer: For time t ≥ 0 hours, let r(t) = 120 1 − e−10t2 represent the speed, in kilometers per hour, at which a ()
car travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is −x 2
modeled by g(x) = 0.05x 1 − e . ()
(a) How many kilometers does the car travel during the first 2 hours?
(b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when
t = 2 hours. Indicate units of measure.
(c) How many liters of gasoline have been used by the car when it reaches a speed of 80 kilometers per
(a)
(b)
hour?
∫ 2 r (t ) dt = 206.370 kilometers 0
dg dg dx dx dt=dx⋅dt; dt=r(t)
2 : {1 : integral 1 : answer
3:{2:useschainrule
1 : answer with units
dg dt
t=2
dg
= dx ⋅r(2)
x=206.370
= (0.050)(120) = 6 liters hour
(c)
Let T be the time at which the car’s speed reaches 80 kilometers per hour.
Then, r(T ) = 80 or T = 0.331453 hours. At time T, the car has gone
x(T ) = ∫ T r (t ) dt = 10.794097 kilometers 0
and has consumed g(x(T )) = 0.537 liters of gasoline.
