Two identical metal spheres, Sphere A and Sphere B

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Two identical metal spheres, Sphere A and Sphere B, are removed from a furnace at the same moment and are placed on separate workbenches in a warehouse whose ambient temperature is held constant at

18^{\circ}\text{C}

. Both spheres have an initial temperature of

340^{\circ}\text{C}

. Because of differences in surface coatings, the proportionality constant governing heat loss is

k_A=0.18\,\text{min}^{-1}

for Sphere A and

k_B=0.12\,\text{min}^{-1}

for Sphere B. Let

T_A(t)

and

T_B(t)

denote the temperatures (in

^{\circ}\text{C}

) of the respective spheres

t

minutes after removal from the furnace. (a) Find explicit expressions for

T_A(t)

and

T_B(t)

. (b) Determine the exact time

t_0

(in minutes) at which the two spheres have the same temperature. Express

t_0

in terms of natural logarithms. (c) Compute the common temperature of the spheres at time

t_0

. (d) For

t>t_0

, identify which sphere cools more rapidly and justify your answer by comparing the magnitudes of the instantaneous rates of change of

T_A

and

T_B

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