Question 466628
We have to find the average value of the velocity, so we integrate v(t) from t = 4 to t = 7, then divide by 3 seconds. This is similar to finding the total distance traveled, then dividing by 3 s to find the average velocity.


Integrating,


*[tex \LARGE \int_{4}^{7} 2t^{\frac{1}{2}} + 4\, dt = \frac{4}{3}t^{\frac{3}{2}} + 4t ]


Evaluate it at t = 7, subtract the value at t = 4:


*[tex \LARGE (\frac{4}{3}7^{\frac{3}{2}} + 4(7)) - (\frac{4}{3}4^{\frac{3}{2}} + 4(3)) = \frac{4}{3} + \frac{28\sqrt{7}}{3}] (meters)


Divide by 3 seconds to obtain


*[tex \LARGE \frac{4}{9} + \frac{28\sqrt{7}}{9}] (meters per second)