The temperature (in degrees Celsius) of a heated metal object immersed in a liquid…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
The temperature T (in degrees Celsius) of a heated metal object immersed in a liquid bath satisfies the differential equation dtdT=−0.25[T−20], where t is measured in minutes and 20∘C is the constant temperature of the bath. At time t=0, the object’s temperature is 120∘C.
(a) Starting at (t0,T0)=(0,120), take three successive time steps of length Δt=0.5 minutes, each time adjusting T by (slope at the left end of the subinterval)×Δt. Record the approximate temperature of the object at t=1.5 minutes, rounded to the nearest tenth of a degree.
(b) Interpret the sign of the slope used in each step in part (a) within the physical context of heat exchange between the object and the bath.
(c) The exact solution of the differential equation is T(t)=20+100e−0.25t. Using this formula, compute the true temperature at t=1.5 minutes and determine the absolute error in the approximation from part (a).
(d) Explain why decreasing the step size Δt would be expected to reduce the magnitude of the error, connecting your reasoning to both the differential equation and the numerical procedure.