In a test run, a new train travels along a straight-line track. Data obtained from the speedometer indicate that the velocity of the train at any time t can be described by the velocity function
v(t) = 8t (0 _< t _< 30)
a. Find the position function of the train.
b. Find the position after 3 seconds. (Note: the train starts from the beginning of the track so when t = 0, the integration constant, C = 0.)
position=INT (v(t)dt
=INT 8tdt=4t^2 + C
b. p(3)=36
can you show me the steps for b. pls
Bob, what the heck does that mean? Need more detail in your steps.
To find the position function of the train, we need to integrate the velocity function with respect to time.
a. The velocity function is given as v(t) = 8t.
To find the position function, we need to integrate v(t) with respect to t:
∫ v(t) dt = ∫ 8t dt
Using the power rule to integrate, we obtain:
= 4t^2 + C
Since the train starts from the beginning of the track, when t = 0, the integration constant C = 0. Therefore, the position function becomes:
s(t) = 4t^2
b. To find the position after 3 seconds, we substitute t = 3 into the position function:
s(3) = 4(3)^2
= 4(9)
= 36
Thus, the position of the train after 3 seconds is 36 units along the track.