Math mini-project


Following is an example of a math mini-project. Students were required to use Scientific Notbebook (or other computer algebra system) to explain an application of differential equations.

Used with permission.

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Newton's Law of Cooling

First_order linear differential equations can be used to solve a variety of problems that involve temperature .For example:

  1. a medical examiner can find the time of death in a homicide case,

  2. a chemist can determine the time required for a plastic mixture to cool to a hardening temperature

  3. and an engineer can design the cooling and heating system of a manufacturing facility although distinct;

Each of these problems depends on a basic principle that is used to develop the associated differential equation, we discuss this important law now.

Newton's law of cooling states that the rate at which the temperature $T(t)$ changes in a cooling body is proportional to the differences between the temperature of the body and the constant temperature $Ts$ of the surrounding medium; this situation is represented as the first_order initial _value problem


Where $To$ is the initial temperature of the body and $k$ is the constant of proportionality; we investigate problems involving Newton's law of cooling in the following examples

In the ideal case , the temperature of the surroundings was assumed to be constant;

However ,this does not have to be the case ;for example, consider the problem of heating and cooling a building ;over the span of a twenty-four hour day, the outside temperature varies ;the problem of determining the temperature inside the building ,therefore ,becomes more complicated ;for the meantime,lets's assume that the building has no heating or air conditioning system ;hence the differential equation that should be solved to find the temperature $u(t)$ at time $t$ inside the building is


Where $C(t)$ is a function that describes the outside temperature and $k>0$ is a constant that depends on the insulation of the building; according to this equation ,if $C(t)>u(t)$,then $\frac{du}{dt}>0,$which implies that $u$ increase ;conversely ,if $C(t)<u(t)$,then $\frac{du}{dt}>0$,which means that $u$ decreases.

(a) suppose that during the month of April in Atlanta, Georgia, the outside temperature is given by MATH,$0<t<24$;(Note: this implies that the average value of $C(t)$ is $21$ $C^{0}$ ) determine the temperature in a building that has an initial temperature of $15$ $C^{0}$ if $k=\frac{1}{4}$;

(b)compare this to the temperature in June when the outside temperature is MATHand the initial temperature is $21$ $C^{0}$

solution: (a)the initial_value problem that we must solve in April is


MATH, Exact solution is: MATH

Next I find the Max temperature at $t=15$


From the above calculating and graph ,we can see that the temperature reaches its Max near $t=15$;and its Max temperature is about $28$ $C^{0}$

(b)solve the temperature in June when the outside temperature is MATHand the initial temperature is $21$ $C^{0}$


MATH, Exact solution is: MATH

Next I find the Max temperature at $t=15$


from the above calculate and the graph ,we can see that for the June ,its Max temperature is about

$33$ $C^{0}$

Again the Max temperature is also near $t=15$,and its value is $33$ $C^{0}.$

Differential equations with maple

Author is Martha l.abell/james p.braselton

Class no. in library of NP: QA371.5 D37 A141