7. a delivery van travels along a straight road. during the time interval 0 ≤ t ≤ 30 seconds, the van’s…

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

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

Answer

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