A factory’s production rate (in units per hour) during a six–hour maintenance cycle…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A factory’s production rate P(t) (in units per hour) during a six–hour maintenance cycle is modeled as follows:
• For 0≤t<3, the machine produces at a constant rate of 60 units/h.
• For 3≤t<4, production is halted while parts are replaced, so P(t)=0.
• For 4≤t≤6, inspection reveals that 20 defective units per hour are removed from inventory, so P(t)=−20.
Time t is measured in hours.
(a) Calculate the net change in the factory’s finished-goods inventory over the entire six–hour period. Show your reasoning without performing any symbolic integration.
(b) Describe, in complete sentences, how the sign of your answer in part (a) relates to the behavior of P(t) on the given intervals.
(c) State the units of your result in part (a) and justify your choice using appropriate unit analysis.
(d) Without additional computation, predict whether the *total number of items processed* (produced plus removed) during the six hours is greater than, equal to, or less than ∣the net change∣. Explain your reasoning clearly.