Senior Maths Comes Alive!
by Michael Jennings
This edition of Infinity sees a new segment, which focuses on a section of Senior Mathematics and relates it to applications discussed in the current issue of Infinity. Here we look at simple models using exponential and logarithmic functions from Maths B.It is sometimes useful to consider simplified models of real world problems and then use the results to develop more complex models such as the predator/prey model discussed on page 2, or the bilby conservation model discussed on page 1. For instance, exponential and logarithmic functions are used by scientists, businesses and banks to model population increases and decreases, economic growth, compound interest, and radioactive decay. For more information on exponential functions and Euler's constant, e, see Infinity 19, page 4. For now, let’s look at how exponential functions are used in real life.
Population: The world’s population is growing at about 2% per annum and the current population (time t = 0) is 6 billion. At the time t years in the future, the population will be approximately P(t) = 6 000 000 000 e0.02t. To estimate the world’s population in 5 years time we calculate P(5) = 6 000 000 000 e0.1 = 6 631 025 508. See if you can find figures on the web to use this model to predict Brisbane’s population in 5 years time.
Financial Mathematics: The interest you receive in your bank account can easily be calculated using an exponential function. If you invest $1000 for two years at 5% compounding annually, you use the formula F = P (1 + r)t, where F is the final value, P is the initial amount, r is the interest rate as a decimal and t is time. So, F = 1000 (1 + 0.05)2.
= 1000 × 1.052
= 1102.50
Therefore the interest gained is 1102.50 – 1000 = $102.50.
Alternatively we could calculate how long we need to leave $1000 invested at 5% compounded annually to obtain $2000 so we can go on a holiday; in this case we need to use the log laws to solve 2000 = 1000 × 1.05t, or equivalently 2 = 1.05t. So
t = log 2/log 1.05
= 14.2 years.
This is a long time!
