A particle moves on a line away from its initial position so that after t hours it is s = 6t2 + 2t miles from its initial position. Find the average velocity of the particle over the interval [1, 4]. Include units in your answer.
To find the average velocity of the particle over the interval [1, 4], we need to calculate the change in position and the change in time.
1. First, we calculate the position of the particle at t = 1 and t = 4:
- At t = 1, the position is s(1) = 6(1)^2 + 2(1) = 8 miles.
- At t = 4, the position is s(4) = 6(4)^2 + 2(4) = 104 miles.
2. Next, we calculate the change in position:
- Change in position = s(4) - s(1) = 104 - 8 = 96 miles.
3. Then, we calculate the change in time:
- Change in time = 4 - 1 = 3 hours.
4. Finally, we calculate the average velocity:
Average velocity = Change in position / Change in time = 96 miles / 3 hours = 32 miles/hour.
Therefore, the average velocity of the particle over the interval [1, 4] is 32 miles/hour.