S=6t^3+8t^2+5 -3≤ t≤ -1
Find the body's average velocity for the given time interval.
S(-3) = 6(-27) + 8(9) + 5 = -85
S(-1) = 6(-1) + 8 + 5 = 7
avg velocity = (7 + 85)/(-1 + 3) = 46
To find the body's average velocity for the given time interval, we need to calculate the change in position divided by the change in time.
Given: S = 6t^3 + 8t^2 + 5 for -3 ≤ t ≤ -1
The change in position (∆S) is the difference between the final position and the initial position. In this case, the final time (t_final) is -1, and the initial time (t_initial) is -3.
∆S = S(t_final) - S(t_initial)
To calculate S for a given time, substitute the value of 't' into the equation for S.
S(t) = 6t^3 + 8t^2 + 5
Substitute the initial time, t_initial, and find S(t_initial):
S(t_initial) = 6(-3)^3 + 8(-3)^2 + 5
Calculate S(t_initial) to find the initial position.
Next, substitute the final time, t_final, and find S(t_final):
S(t_fina) = 6(-1)^3 + 8(-1)^2 + 5
Calculate S(t_final) to find the final position.
Finally, calculate the change in position (∆S) by subtracting S(t_initial) from S(t_final).
∆S = S(t_final) - S(t_initial)
Now, calculate the change in time (∆t) by subtracting t_initial from t_final.
∆t = t_final - t_initial
Finally, calculate the average velocity by dividing ∆S by ∆t:
Average velocity = ∆S / ∆t