# HSC Extension 1 and 2 Mathematics/3-Unit/HSC/Applications of calculus to the physical world

## Exponential Growth and Decay

2 unit course

The exponential function can be used to show the growth or decay of a given variable, including the growth or decay of population in a city, the heating or cooling of a body, radioactive decay of radioisotopes in nuclear chemistry, and amount of bacteria in a culture.

The exponential growth and decay formula is $N = N$0ekt

where:
$N$0
is the first value of N (where $t = 0$)
$t$
represents time in given units (seconds, hours, days, years, etc.)
$e$ is the exponential constant ($e = 2.718281828...$), and
$k$ is the growth ($k +ve$) or decay($k -ve$) constant.

Differentiation can be used to show that the rate of change (with respect to time, $t$) of $N$ is proportional (∞) to $N$. if:
$N = N$0ekt,
then the Derivative of $N$ can be show as:
dN $=kN$0ekt
dt
$=kN$, substituting $N = N$0ekt.

(note the derivative of e is the variable $k$ of the power of e times $e^{kx}$ and $N, t$ are constant.)

## 3 Unit applications

not yet complete

The variable of a given application can be proportionate to the difference between the variable and a constant. An example of this is the internal cooling of a body as it adjusts to the external room temperature.

dN = $k(N - P)$
dt
$= kN$0ekt$- kP$

where $P$ = the external constant (e.g., the external room temperature)

using natural logarithms, $log$e$x$, we can find any variable when given certain information.
Example:
A cup of boiling water is initially $100$oC. The external room temperature is $24$oC. after 10 minutes, the temperature of the water is $74$oC. find
(i) k
(ii)how many minutes it takes for the temperature to equal 30 degrees.

(i)$74=24-100$e10k
$50=-100$e10k

$\frac {50} {-100} = e^{10k}$

$log$e$1 - log$e$(-2) = 10k$

$k = \frac{log_e 1 - log_e(-2)}{2}$

= .34567359... (store in memory)

(ii) 30=24-100e^(.34657359t)

incomplete 10th august '08