A hot beverage cools on a countertop

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A hot beverage cools on a countertop. Its temperature

T

(in degrees Celsius) for

0\text{ min} \le t \le 10\text{ min}

satisfies

T'(t)=k(t)

, where the cooling/heating rate

k

(in

^\text{°C/min}

) is given by

k(t)=\begin{cases} -3e^{-0.5t}+0.5, & 0\le t<4,\\ 2, & 4\le t<6,\\ -1.5\sin\bigl(\tfrac{\pi}{2}(t-6)\bigr), & 6\le t\le 10. \end{cases}

t=0

the beverage is at

80^{\circ}\text{C}

. (1) Part 1: (a) Solve

k(t)=0

0\le t\le 10

to identify every time at which the temperature is momentarily not changing. (2) Part 2: (b) Write an expression for

T(10)-T(0)

as a sum of definite integrals, splitting at every time you found in part (a) and at each point where

k

changes its defining formula. (3) Part 3: (c) Evaluate the integrals from part (b) to determine the temperature of the beverage at

t=10

minutes. Express exact values; do not approximate. (4) Part 4: (d) The beverage is considered ready to drink when its temperature first drops to

65^{\circ}\text{C}

. Use the Fundamental Theorem of Calculus and appropriate reasoning to show that this occurs at a time

t

satisfying

4<t<6

. Then, using one additional definite integral you set up, determine the value of

t

to the nearest tenth of a minute.