# Exponential Modelling

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Exponential functions are used to model many physical occurances including growth of cells, population growth, continuously compounded interest, radioactive decay and the rate of cooling (see Newton's Law of Cooling).

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Let A be the quantity at time t. Then $A = A_0e^{kt}$,

- where
**A**is the initial quantity (a constant)_{0} - where
**k**is the rate constant of the equation. - where $t \geqslant 0$

**Note:**

… … Growth: **k > 0**

… … Decay: **k < 0**

**Note:**

- Most functions involved with modelling have a domain of $t \geqslant 0$
- and many have a co-domain of $A \geqslant 0$
- so take care to only draw the appropriate part of the graph (usually 1st quadrant)

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#### Example 1

The population of wombats in an area is given by:

… … $W = 100e^{0.03t}$

… … … where **W** is the number of wombats

… … … t is the time in years after 1 January, 1998.

… **a)** … Sketch **W** against **t** for $t \in [0, \; 30]$

… **b)** … Find the time taken for the population of wombats to double.

… … … State the year and the week.

**Solution**

… **a)** … Sketch **W** against **t** for $t \in [0, \; 30]$

… … … Note that **t** is always positive and **W** is always positive

… … … so only draw the 1st Quadrant.

… … … Note the asymptote plays no part in this portion of the graph

… … … so it needn't be marked in.

… … … Note the labelling of the axes is **W** and **t** (not x and y)

… … … Note that significant points are identified and labelled

… … … (in this case the endpoints of the domain).

… … … Note that the initial number of Wombats is $W_0 = 100$

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… **b)** … Find the time taken for the population of wombats to double.

… … … State the year and the week.

… … … This can be found algebraically on the calculator

… … … by solving for **t** when $W = 200$

**OR**

… … … To find graphically, sketch W = 200 on the same graph

… … … and locate the point of intersection.

… … … Point of intersection at $t = 23.1049$

… … … 23 years after 1998 is 2021

… … … $0.1049 \times 52 = 5.4548$

… … … ie population has not yet doubled by the 5th week

… … … so population has doubled by the 6th week of 20121

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