Question

1. a delivery van travels along a straight road. during the time interval 0 ≤ t ≤ 30 seconds, the van’s velocity in feet per second is a continuous function. use the table below to find the minimum number of times that the van must have been stopped. justify your answer.

t (sec)	0	5	7	12	18	22	30
v(t) (ft/sec)	-28	-60	-15	8	24	-4	10

Explanation:

Step1: Recall the condition for stopping

The van is stopped when \(V(t)=0\). Since \(V(t)\) is a continuous function, by the Intermediate - Value Theorem, if \(V(t)\) changes sign (from negative to positive or from positive to negative) over an interval, then \(V(t)\) must be \(0\) at some point in that interval.

Step2: Check sign - changes

At \(t = 0\), \(V(0)=- 28\) (negative), at \(t = 5\), \(V(5)=-60\) (negative), at \(t = 7\), \(V(7)=-15\) (negative), at \(t = 12\), \(V(12)=8\) (positive). Since \(V(t)\) changes sign from negative to positive between \(t = 7\) and \(t = 12\), \(V(t)=0\) at least once in the interval \((7,12)\).
At \(t = 18\), \(V(18)=24\) (positive), at \(t = 22\), \(V(22)=-4\) (negative). Since \(V(t)\) changes sign from positive to negative between \(t = 18\) and \(t = 22\), \(V(t)=0\) at least once in the interval \((18,22)\).

Answer:

The van must have been stopped at least 2 times.