A metal sphere at a temperature of 120 °C is placed in a room that is maintained at a…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A metal sphere at a temperature of 120 °C is placed in a room that is maintained at a constant temperature of 25 °C. Let T(t) denote the temperature of the sphere (in °C) after t minutes.
(i) Write the differential equation that governs T(t) under the usual physical assumption relating the rate of temperature change to the temperature difference.
(ii) Explain the meaning and sign of the proportionality constant appearing in this equation.
(iii) Without solving the differential equation, describe what happens to T(t) as t→∞ and justify your answer.
(iv) Sketch a qualitative graph of T(t) against t and clearly label the initial point, the limiting temperature and the general shape of the curve.