During a 40-minute baking period

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

During a 40-minute baking period, the internal temperature

T

(in degrees Celsius) of a loaf of bread is modeled by a differentiable function on the interval

0 \le t \le 40

, where

t

is measured in minutes. The loaf is placed in the oven at

t = 0

with an internal temperature of

24^{\circ}\text{C}

and is removed at

t = 40

with an internal temperature of

208^{\circ}\text{C}

. Food-safety guidelines require that the rate at which the internal temperature increases must never exceed

20^{\circ}\text{C}

per minute. (a) Determine the average rate of change of the loaf’s internal temperature over the interval

0 \le t \le 40

. Include units in your answer. (b) Explain why there must be at least one time

c

with

0 < c < 40

for which

T'(c)

equals the average rate found in part (a). (c) Using your result from part (a), evaluate whether the available information alone guarantees that the loaf met the food-safety guideline during the entire baking period. Support your answer with a clear mathematical argument. (d) Suppose additionally that temperature probes inserted at

t = 10

and

t = 30

recorded instantaneous heating rates of

18^{\circ}\text{C}

/min and

22^{\circ}\text{C}

/min, respectively. What conclusions can you draw about compliance with the guideline? Justify your answer.

Related theory

Browse the problem library