A metal sphere with temperature (in degrees Celsius) is placed in a liquid bath…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A metal sphere with temperature T(t) (in degrees Celsius) is placed in a liquid bath maintained at a constant temperature Tb. Experimental data suggest that the instantaneous rate of change of the sphere’s temperature is proportional to the temperature difference T(t)−Tb.
Assume T(t) is differentiable for t≥0, where t is measured in minutes.
(a) Formulate a differential equation relating dtdT and T(t) that incorporates an appropriate parameter. Do not determine a numerical value for the parameter.
(b) Using dimensional analysis, determine the physical units of the parameter you introduced in part (a). Show your reasoning.
(c) Explain why the parameter in part (a) must be negative for the sphere to cool toward the bath temperature, and describe how its magnitude affects the rate at which the sphere approaches thermal equilibrium.