A model for the ambient temperature (in °C) inside a greenhouse on a spring day is…

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

Problem

A model for the ambient temperature

T

(in °C) inside a greenhouse on a spring day is given by

T(t) = \begin{cases} 0.8t + 12, & 0 \le t < 12, \\ Ae^{-0.3(t-12)} + B, & t \ge 12, \end{cases}

where

t

is the number of hours after midnight, and

A

and

B

are real constants. (1) Part 1: (a) Determine

\displaystyle \lim_{t\to 12^-} T(t)

and

\displaystyle \lim_{t\to 12^+} T(t)

in terms of

A

and

B

. Show all algebra involved. (2) Part 2: (b) The physical situation requires the temperature model to be continuous at noon (

t=12

). Derive the relationship between

A

and

B

imposed by this condition. (3) Part 3: (c) Historical data indicate that the temperature at midnight (

t=0

) was

12^{\circ}\text{C}

and at noon the measured temperature was

21.6^{\circ}\text{C}

. Use this information together with the condition found in part (b) to determine the numerical values of

A

and

B

. Show your complete solution process. (4) Part 4: (d) With the values of

A

and

B

obtained in part (c), discuss whether any discontinuities remain in the model for

0\le t <24

. Identify the type of each discontinuity, if any, and support your conclusions with appropriate limit calculations.