A model for the internal temperature (in degrees Celsius) of a chemical reactor…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A model for the internal temperature

T

(in degrees Celsius) of a chemical reactor during the first hour of operation is given by the piecewise function

T(t)=\begin{cases} 25e^{0.04t}, & 0 \le t<15,\\[6pt] ct+50, & 15\le t\le 60, \end{cases}

where

t

is measured in minutes and

c

is a real constant. The engineers require the model to be continuous over the closed interval

[0,60]

minutes. (a) Determine the value of

c

that satisfies the continuity requirement. Explicitly evaluate any limits needed in your argument. (b) Using the value of

c

obtained in part (a), interpret the slope of the linear expression

ct+50

in the context of the reactor’s temperature profile. Include correct units in your explanation. (c) Explain why the function you obtained in part (a) is automatically continuous on

[0,15)

and

(15,60]

, and why checking continuity only at

t=15

is sufficient for the entire interval.

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