An automobile accelerates from rest at 1 + 3 * sqrt (t) mph/sec for 9 seconds.

how far does it travel in those 9 sec?

Writing the acceleration as a function of time:

a (t) = 1 + 3√t

Integrating acceleration, we obtain velocity:

v (t) = t + 2 (t) ^ (3/2) + c;

object at rest so velocity at t = 0 is 0 so c = 0.

v (t) = t + 2 (t) ^ (3/2)

Integrating velocity to obtain an equation for displacement:

d (t) = t²/2 + 4/5 t^ (5/2) + c

Applying limits from t = 0 to t = 9

d = 9²/2 + 4/5 9^ (5/2)

d = 234.9 m