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.
I started by taking the derivative of s(t) but am not sure how the interval fits into that. Any help or hint to solve is greatly appreciated!
Since you want the average velocity, you don't even need the derivative
when t = 1, s = 6(1) + 2(1) = 8
when t = 4, s = 6(16) + 2(4) = 104
average speed = total distance/total time
= (104-8)/(4-1)
= 32 mph
To find the average velocity of the particle over the interval [1, 4], we need to find the change in position (distance) and divide it by the change in time.
First, let's find the position at time t = 4:
s(4) = 6(4^2) + 2(4) = 96 + 8 = 104 miles
Now, let's find the position at time t = 1:
s(1) = 6(1^2) + 2(1) = 6 + 2 = 8 miles
The change in position over the interval [1, 4] is:
Δs = s(4) - s(1) = 104 - 8 = 96 miles
The change in time over the interval [1, 4] is:
Δt = 4 - 1 = 3 hours
Finally, we can calculate the average velocity:
average velocity = Δs / Δt = 96 miles / 3 hours
The average velocity of the particle over the interval [1, 4] is 32 miles per hour.
To find the average velocity of the particle over the interval [1, 4], we can use the definition of average velocity, which is the change in position divided by the change in time.
The change in position over the interval [1, 4] can be calculated by subtracting the initial position at t=1 from the final position at t=4.
To find the initial position at t=1, we substitute t=1 into the position equation s(t) = 6t^2 + 2t:
s(1) = 6(1)^2 + 2(1) = 6 + 2 = 8 miles.
To find the final position at t=4, we substitute t=4 into the position equation:
s(4) = 6(4)^2 + 2(4) = 6(16) + 8 = 96 + 8 = 104 miles.
So, the change in position over the interval [1, 4] is:
Change in position = 104 - 8 = 96 miles.
The change in time over the interval [1, 4] is 4 - 1 = 3 hours.
Therefore, the average velocity of the particle over the interval [1, 4] is given by:
Average velocity = Change in position / Change in time = 96 miles / 3 hours = 32 miles/hour.
So, the average velocity of the particle over the interval [1, 4] is 32 miles/hour.