A thermostat is programmed to warm a warehouse beginning at midnight

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A thermostat is programmed to warm a warehouse beginning at midnight. For

0 \le t \le 8

, where

t

is measured in hours after midnight and

T(t)

is the air temperature (in °F), the temperature model is

T(t)=\begin{cases} 60+4t, & 0\le t \le 3,\\[6pt] A\,e^{k(t-3)}+B, & 3< t \le 8. \end{cases}

t=8

, sensors record

T(8)=75

°F, and engineers insist the model be continuous on the entire interval

[0,8]

so that calculus tools may later be applied to analyze rates of change. (a) Determine the values of the constants

A

and

B

that guarantee

T(t)

is continuous at

t=3

. (b) The parameter

k

will later be used to fit the exponential portion to additional data, but continuity must hold for every possible

k

. Verify algebraically that, once the constants from part (a) are chosen,

T(t)

is continuous on the closed interval

[0,8]

regardless of the value of

k

. (c) The warehouse manager needs to know whether there is a time during

0\le t\le 8

when the air temperature is exactly 70 °F. Using your work from part (b) and appropriate theorems, justify that such a time must exist. (d) Interpret the conclusion in part (c) in the context of the warehouse, specifying the practical significance for the manager’s planning.

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