An object at temperature (in °C) cools in a room held at a constant 20 °C
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
An object at temperature T(t) (in °C) cools in a room held at a constant 20 °C. The temperature satisfies dtdT=−0.1(T−20) for t≥0, where t is measured in minutes.
(a) On the grid of points with t-coordinates 0,1,2,3,4,5 and T-coordinates 10,20,30,40, compute the value of dtdT at each point.
(b) Use the slopes from part (a) to sketch a slope field covering 0≤t≤5 and 10≤T≤40. Draw a representative sample of line segments so the overall pattern is clear.
(c) A solution curve passes through (0,30). Using only the slope field, estimate the object’s temperature at t=5 minutes and explain the steps in your estimation.
(d) Interpret in everyday language what the slope field suggests about how quickly the object cools when it is much warmer than the room versus when it is close to room temperature.