A manufacturing process adds or removes heat from a metal rod according to a rate (in…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A manufacturing process adds or removes heat from a metal rod according to a rate (in joules per second) described verbally as follows.
• From t=0 to t=2 seconds, heat is added at a constant rate of 5 J/s.
• From t=2 to t=5 seconds, heat is removed at a rate that decreases linearly from 0 J/s at t=2 to −6 J/s at t=5.
• From t=5 to t=7 seconds, heat is added again at a constant 4 J/s.
At t=0, the rod contains 60 joules of thermal energy.
(a) Express the rate function H(t) as a piecewise-defined analytic function on 0≤t≤7.
(b) Compute the net change in thermal energy of the rod on 0≤t≤7 seconds.
(c) Determine the total amount of heat transferred (added plus removed) during the same time period.
(d) Find the exact thermal energy content of the rod at t=7 seconds and interpret your answer in the context of the scenario.