Question

A toy car accelerates from rest traveling in a straight line. Let \(d=s(t)\) represent the distance a toy car has moved away from its initial position (in feet) and let \(t\) represent the number of seconds elapsed since the toy car started moving. Values of \(d=s(t)\) for values of \(t = 1.5\) to \(t = 2\) are provided in the table below.

   t    d
 1.5 22.5
 1.6 25.6
 1.7 28.9
 1.8 32.4
 1.9 36.1
 2.0 40.0

Select each of the true statements from the following:

Answerlist

  • \(s(1.7)=28.9\)

  • The value \(t^*\) in the equation \(s(t^*)=28.9\) is the time (in minutes) it takes the car to travel \(28.9\) feet.

  • \(s(1.7)=1.7\)

  • The car travels \(1.7\) feet over the first \(28.9\) seconds of motion.

  • An approximation of the speed of the toy car \(1.7\) seconds after starting is \[ \frac{25.6 - 28.9}{0.1} \].

  • An overestimate of the speed of the toy car \(1.7\) seconds after starting is \[ \frac{28.9 - 25.6}{0.1} \].

  • An underestimate of the speed of the toy car \(1.7\) seconds after starting is \[ \frac{28.9 - 22.5}{0.2} \].

Solution

Answerlist

  • True. Looking at the table we can see that the value of \(d\) that corresponds to \(1.7\) is \(28.9\). Therefore, \(s(1.7)=28.9\).

  • False. The units of \(t\) are seconds, not minutes.

  • False. The \(s\) functions produces values of \(d\). The number \(1.7\) is not a value of \(d\).

  • False. Over the interval \([0,1.7]\) the car travels \(d=s(1.7) = 28.9\) feet.

  • False. The calculation \[ \frac{25.6 - 28.9}{0.1} \] is negative but the speed should be positive since the distances are increasing.

  • False. Since \(s\) is an increasing function and the car is accelerating, an overestimate of the speed of the toy car \(1.7\) seconds after starting is \[ \frac{32.4 - 28.9}{0.1} \]. An underestimate of this speed \[ \frac{28.9 - 25.6}{0.1} \]

  • False. Since \(s\) is an increasing function and the car is accelerating, an overestimate of the speed of the toy car \(1.7\) seconds after starting is \[ \frac{32.4 - 28.9}{0.1} \]. An underestimate of this speed \[ \frac{28.9 - 25.6}{0.1} \]

Meta-information

extype: mchoice exsolution: 1000000 exname: Approximating Instantaneous Rates of Change