Let a function represent the vertical acceleration (in meters per second squared) of…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let a function g represent the vertical acceleration (in meters per second squared) of an elevator over the time interval 0≤t≤12 seconds:
g(t)=⎩⎨⎧0.8t,2.4−0.8t,−sin(3π(t−6)),0≤t<3,3≤t<6,6≤t≤12.
At t=0 the elevator is at rest (v(0)=0) and is located 5 meters below ground level (s(0)=−5). Let v denote velocity and s denote position measured from ground level.
(a) Determine all values of t in 0≤t≤12 for which g(t)=0. Arrange these times in increasing order.
(b) Express the velocity v(12) as a sum of definite integrals involving g(t), clearly indicating each limit of integration. Then compute v(12).
(c) Write an expression, in terms of definite integrals, for the net change in position s(12)−s(0). Evaluate this expression exactly.
(d) Calculate the total distance traveled by the elevator during 0≤t≤12 seconds. Provide a clear justification for how you partitioned the interval and how you accounted for direction changes.
(e) Using your results, compute the average value of the velocity of the elevator on 0≤t≤12 and interpret this average value in context.