A glass container holding a chemical solution is heated to and then transferred…

MATHEMATICS · Subjective · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A glass container holding a chemical solution is heated to

95^{\circ}\mathrm{C}

and then transferred immediately to a cold room kept at

5^{\circ}\mathrm{C}

. During the first

8\,\text{min}

it is accidentally left in the laboratory where the ambient temperature is

20^{\circ}\mathrm{C}

. At the end of these

8\,\text{min}

its temperature is measured to be

68^{\circ}\mathrm{C}

, after which it is placed in the cold room. Seven minutes after being placed in the cold room its temperature is observed to be

20^{\circ}\mathrm{C}

. Assume that each environment exerts its own constant cooling effect with its own (possibly different) cooling constant. Let

k_1

be the cooling constant while the solution is in the laboratory and

k_2

be the cooling constant in the cold room. (a) Show that the laboratory reading leads to a numerical value of

k_1

(express your answer correct to three decimal places in

\text{min}^{-1}

). (b) Using the temperature at the end of the laboratory period as the new initial temperature for the cold room phase, obtain the value of

k_2

. (c) How long, measured from the instant it entered the cold room, will the solution take to reach

10^{\circ}\mathrm{C}