A car drives down a road in such a way that its velocity ( in m/s) at time t (seconds) is v(t)=3(t^(1/2))+4. Find the car's average velocity (in m/s) between t=4 and t=6.
To find the average velocity of the car between t=4 and t=6, we need to find the total displacement of the car during this time interval and divide it by the total time.
The velocity of the car at time t is given by the equation v(t) = 3(t^(1/2)) + 4.
To find the displacement of the car between t=4 and t=6, we need to find the difference in position of the car at these two times.
We can do this by integrating the velocity function. Taking the integral of v(t) with respect to t will give us the position function of the car.
Let's integrate v(t) to find the position function of the car:
∫ v(t) dt = ∫ (3(t^(1/2)) + 4) dt
= ∫ (3t^(1/2)) dt + ∫ 4 dt
Integrating 3t^(1/2) with respect to t gives (2/3)(t^(3/2)):
∫ (3t^(1/2)) dt = (2/3)(t^(3/2)) + C
Integrating 4 with respect to t gives 4t:
∫ 4 dt = 4t + C
So the position function of the car is given by p(t) = (2/3)(t^(3/2)) + 4t + C, where C is the constant of integration.
Now we can find the displacement of the car between t=4 and t=6 by evaluating the position function at these two times:
Displacement = p(6) - p(4)
= [(2/3)(6^(3/2)) + 4(6) + C] - [(2/3)(4^(3/2)) + 4(4) + C]
= (2/3)(6^(3/2)) + 4(6) - (2/3)(4^(3/2)) - 4(4)
= (2/3)(36^(1/2)) + 24 - (2/3)(16^(1/2)) - 16
= (2/3)(6) + 24 - (2/3)(4) - 16
Simplifying further:
Displacement = (12/3) + 24 - (8/3) - 16
= 4 + 24 - (8/3) - 16
= 28 - (8/3) - 16
= 12 - (8/3)
= (36/3) - (8/3)
= 28/3
Now, we can find the average velocity by dividing the displacement by the total time:
Average velocity = Displacement / Total time
= (28/3) / (6 - 4)
= (28/3) / 2
= 28/6
= 14/3
Therefore, the average velocity of the car between t=4 and t=6 is 14/3 m/s.