The temperature in degrees Celsius of a cooling liquid obeys the differential equation
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
The temperature T(t) in degrees Celsius of a cooling liquid obeys the differential equation dtdT=−k(T−20), where the ambient temperature is 20∘C and k is a positive constant. At time t=0 minutes the liquid is at 90∘C. A laboratory technician recorded successive numerical approximations of T(8), refining the uniform step size by halving it at each stage. The data are shown below.
step size h (min)
approximation to T(8) (°C)
2.0
42.37
1.0
38.96
0.5
37.27
0.25
36.43
Because of computational resource limits, no more than 32 total time steps may be performed.
(a) Compute the absolute error for each approximation if the exact temperature at t=8 minutes is 36.02∘C.
(b) Decide whether any approximation already satisfies a maximum allowable error of 0.3∘C. If none does, propose the next step size you would test, keeping the step-limit constraint in mind.
(c) Provide quantitative evidence, based on the pattern of errors in part (a), to justify your proposal in part (b). Your justification should address both the expected improvement in accuracy and the total number of time steps required.