A hot liquid cools in a room kept at a constant temperature of

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

A hot liquid cools in a room kept at a constant temperature of 20∘C20^{\circ}\text{C}. Its temperature TT (in degrees Celsius) satisfies the differential equation dTdt=−0.1 (T−20)\dfrac{dT}{dt}=-0.1\,(T-20), where tt is measured in minutes. The liquid is initially at 80∘C80^{\circ}\text{C} when t=0t=0. For quality control, an inspector uses the following monitoring procedure: beginning at (0,80)(0,80), successive checkpoints are determined by moving forward Δt=2\Delta t=2 minutes in time and adjusting the temperature by ΔT=(dTdt)(t,T) Δt\Delta T=(\tfrac{dT}{dt})_{(t,T)}\,\Delta t computed at the current checkpoint. (a) Determine the temperature estimates at t=2t=2, t=4t=4, and t=6t=6 minutes produced by the inspector’s procedure. (b) A shipment can proceed only after the liquid cools below 40∘C40^{\circ}\text{C}. Based on the checkpoint data from part (a), state whether shipping would be permitted at t=6t=6 minutes and explain your answer. (c) Without performing further numerical work, discuss one specific limitation of the inspector’s procedure that could lead to an incorrect shipping decision in part (b). (d) Describe an adjustment to the procedure that would address the limitation identified in part (c) and explain why the adjustment would be effective.

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