03Iexplogmodelling

toc = Exponential and logarithmic modelling =

**Exponential functions** are used to model many physical occurrences such as: growth of cells, population growth, continuously compounded interest, radioactive decay, Newtons's law of cooling.

Let **//A//** be the quantity at time t. Then: ... ... //** A = A **//** 0 **//** e kt **//,

where **//A 0 //** is a constant and represents the ** initial ** value of A. (ie at t = 0)

The number **//k//** is the rate constant of the equation.
 * Growth: **//k//** > 0 {increasing over time}
 * Decay: **//k//** < 0 {decreasing over time}

** Example 1 **
The population of a town was 8,000 at the beginning of 1992 and 15,000 at the end of 1999. Assume the rate of growth is exponential. ... ... (a) Find the population at the end of 2001. ... ... (b) In what year will the population be double that of 1999?

__** Solution: **__

Let
 * **//P//** //=// population of town
 * **//t//** //=// time in years (measured from January 1, 1992)

Exponential population growth can be modelled by: **//P = P 0 e kt //**.

At the beginning of 1992,
 * t = 0
 * P = 8000

... ... 8000 = P 0 e 0

... ... P 0 = 8000 {Showing that P 0 is the __initial__ population}

... ... //**P = 8000e kt **//

To find k, use the other set of information we were given.

At the end of 1999,
 * t = 8
 * P = 15,000

... ... 15000 = 8000e 8k

math . \qquad e^{8k}=\dfrac{15}{8} \;\; \iff \;\; \log_e \left( \dfrac{15}{8} \right)=8k math

math . \qquad k=\dfrac{1}{8} \log_e \left( \dfrac{15}{8} \right) math

. ... ... k = 0.07858

The rate of increase of the populaton is 7.9% per annum.

Exponential growth equation is: ... ... P = 8000e 0.07858t


 * (keep longer k value in your calculator for later calculations, but don't write too many decimal places) **


 * (a) ** At the end of 2001, t = 10

... ... P = 8000e 0.07858×10

... ... P =17553

The town's population at the end of 2001 is approximately 17,550.

When does P = 30,000? math \\ . \qquad 30000=8000e^{0.07858 \times t} \\. \\ . \qquad e^{0.07858 \times t}=\dfrac{30}{8} \\. \\ . \qquad \log_e{\frac{30}{8}}=0.07858 \times t \\. \\ . \qquad t=16.82 math
 * (b) **

The population reaches 30,000 in the 17th year, that is during 2008.

Modelling on the Calculator
This question could have been answered using the CAS calculator.

To find the original equation, use __exponential regression__:

Go to the statistical data entry page and enter the information we are given:
 * (0, 8000) and
 * (8, 15000)

Select **Exponential Regression** and you will get the equation: math . \qquad y=8000e^{0.078576x} math

Make sure this equation is in the graphing section (y1 = ...) then graph the equation over a suitable domain.

To find the population after 10 years, go to trace and enter x = 10

To find when the population reaches 30,000: Draw in a second equation: y2 = 300000 Find the point where the two lines intersect.

From the calculator, population reaches 30,000 when x = 16.82 (ie during 2008)

Modelling with Logarithmic Functions

 * Logarithmic functions** are used to model physical applications such as: magnitude of earthquakes, intensity of sound, acidity of a solution.

** Example 2 **
A solute is being dissolved in water. The concentration of the solute in the solution is modelled by **//C = log e k(t// – //b)//**
 * where **//C//** = concentration in moles per litre (M)
 * and **//t//** = mixing time (seconds).

If the concetration is 0.4 M after 5 seconds and 1.2 M after 20 seconds, find: ... ... (a) The values of k and b correct to 3 decimal places ... ... (b) How long does the solution need to be mixed to completely dissolve?

... ... Assume the solute has completely dissoved when the concentration reaches 2 M.

__** Solution: **__

... ... C = log e k(t – b)
 * (a) **

... ... When t = 5, C = 0.4 ... ... 0.4 = log e k(5 – b). . . . ** (1) **

... ... When t = 20, C = 1.2 ... ... 1.2 = log e k(20 – b). . . . ** (2) **

Use the CAS calculator to solve simultaneously:

... ... k = 0.122, ... ... b = –7.239

... ... C = log e 0.122(t + 7.239)
 * (b) **

Need to find t when C = 2

... ... 2 = log e 0.122(t + 7.239)

Using the CAS calculator we get ... ... t = 53.327 seconds

The solute will completely dissolve after 54 seconds of mixing.

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