Question

Anne Kates house is 20c. She had just made a fresh cup of tea (100c). Five minutes after she made her mad scientist nephew came in,stuck a thermometer in the cup and announced that the tea was now only 70c. Now she won’t drink it because it isn’t piping hot anymore.Write an equation that models this problem.

Answer 1

We define the following variables and functions:

• t = time,

,

• T(t) = temperature as a function of time.

From the statement, we know that:

• the initial temperature is T(0) = 100°C,

,

• the temperature at time t = 5 minutes is T(5) = 70°C.

If we consider a linear model to describe this problem, we have the general linear equation:

$T(t)=m\cdot t+T(0)\text{.}$

Where m is the rate of change of the temperature. Replacing the data above for t = 5 min, we have:

$T(5)=m\cdot5\min +100\degree C=70\degree C.$

Solving for m, we get:

$\begin{gathered} m\cdot5\min =70\degree C-100\degree C, \\ m\cdot5\min =-30\degree C, \\ m=\frac{-30\degree C}{5\min}=-6\cdot\frac{\degree C}{\min}\text{.} \end{gathered}$

Replacing this value and T(0) = 100°C in the general equation, we get:

$T(t)=-6\cdot\frac{\degree C}{\min}\cdot t+100\degree C\text{.}$

Answer

The equation that model this problem is:

$T(t)=-6\cdot\frac{\degree C}{\min}\cdot t+100\degree C\text{.}$