04Hmodelling

= Exponential Modelling =

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

Let A be the quantity at time t.

Then ... ... A = A 0 e kt ,
 * where A 0 is the initial quantity (a constant)
 * where k is the rate constant of the equation.
 * where t __>__ 0


 * Note: **
 * Growth: k > 0
 * Decay: k < 0


 * Note: **
 * __**Most**__ functions involved with modelling have a **__domain__** of t __>__ 0
 * and __**many**__ have a __**co-domain**__ of A __>__ 0
 * so take care to only draw that part of the graph (usually 1st quadrant)


 * 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 Î [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

This can be found algebraically.
 * (b) ** Find the time taken for the population of wombats to double. State the year and the week.

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 * 52 = 5 ie population has doubled by the 5th week of 2021.

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