A manufacturer’s quality–control team models the vertical displacement (in…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A manufacturer’s quality–control team models the vertical displacement (in millimeters) of two conveyor belts above a reference height with the functions
u(t)=t(3−t)andv(t)=1−2(t−1)2,0≤t≤3,
where t is measured in seconds. The belts share the same horizontal span and occasionally touch, trapping objects between them.
(a) Determine all times in [0,3] at which the conveyor belts are at the same height. Show analytical work leading to your solutions.
(b) Engineers wish to calculate the total cross–sectional area (in square millimeters) of the region between the two belts for 0≤t≤3. Provide a single definite integral whose value is that area, then evaluate it exactly.
(c) The belts will damage fragile items whenever the region from part (b) has area exceeding 65 square millimeters. Should the belts be stopped? Fully justify your conclusion with a quantitative argument.