Final answer:
To find who is ahead after 5 hours, integrate the velocity functions of Abe and Bob over the interval from t=0 to t=5, and compare the distances traveled. The runner with the greater distance will be the one ahead.
Step-by-step explanation:
The student asks who will be ahead after t=5 hours if Abe and Bob race starting at the same time and place, with Abe running at velocity u(t)=4/(t+1) and Bob running at velocity v(t)=4e⁻¹¹/2 mi/hr. To determine who is ahead, we calculate the distance each runner has traveled by the time t=5 hours by integrating their velocity functions with respect to time.
For Abe: distance = ∫ u(t) dt = ∫ (4/(t+1)) dt from t=0 to t=5.
For Bob: distance = ∫ v(t) dt = ∫ (4e⁻¹¹/2) dt from t=0 to t=5.
After evaluating these integrals, we obtain the total distance ran by each runner. The runner with the larger distance will be ahead after 5 hours.