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 stopped - state

The van is stopped when \(V(t)=0\). Since \(V(t)\) is a continuous function, by the Intermediate - Value Theorem, if \(V(a)\) and \(V(b)\) have opposite signs, then there must be at least one \(c\in(a,b)\) such that \(V(c) = 0\).

Step2: Check sign - changes of \(V(t)\)

1. Between \(t = 7\) and \(t = 12\), \(V(7)=- 15\) (negative) and \(V(12)=8\) (positive). So, there is at least one time \(t_1\in(7,12)\) such that \(V(t_1)=0\).
2. Between \(t = 18\) and \(t = 22\), \(V(18)=24\) (positive) and \(V(22)=-4\) (negative). So, there is at least one time \(t_2\in(18,22)\) such that \(V(t_2)=0\).

Answer:

2