A question on a test asks students to find the speed at which a car travels. The graph shows a proportional relationship between the distance traveled in miles and time in hours. billy incorrectly says that the speed of the car is 1/62 mile per hour. What is the speed of the car? What error might billy have made?

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.