A cup of coffee cools in a room whose ambient temperature is

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

25^{\circ}\text{C}

. The temperature

T

of the coffee (in degrees Celsius) satisfies

\dfrac{dT}{dt}=-0.07\,(T-25)

, where

t

is measured in minutes. At time

t=0

, the coffee temperature is

90^{\circ}\text C

. A numerical plan is proposed to estimate the time at which the coffee first cools to

60^{\circ}\text C

. The first subinterval length is chosen to be

h_1=2

minutes. After completing the

k^{\text{th}}

step, the subinterval length is adjusted as follows: • If the magnitude of the temperature drop on that step exceeds

4^{\circ}\text C

, the next subinterval length is halved. • If the magnitude of the temperature drop is less than

2^{\circ}\text C

, the next subinterval length is doubled, but may not exceed

4

minutes. (a) Formulate a clear criterion, expressed in symbolic form, that determines

h_{k+1}

from

h_k

T_{k-1}

, and

T_k

. (b) Beginning with

T_0=90

t_0=0

, carry out the first three steps of the procedure, listing

t_k

h_k

, and

T_k

for

k=1,2,3

. (c) Continue the adaptive process (additional steps may be summarized rather than shown in full detail) until the temperature first falls to or below

60^{\circ}\text C

. Report the estimated cooling time and give a brief justification of the accuracy of your estimate based on the final temperature drop observed. (d) Discuss two advantages and one potential drawback of the adaptive subinterval strategy used here when compared with employing a fixed subinterval length throughout the cooling process.

Related theory

Browse the problem library