10.2 Exponential Growth/Decay of a Practical Phenomenon through its Graph

10.2 Exponential Growth/Decay of a Practical Phenomenon through its Graph

Exponential growth and decay are widely observed in real-world phenomenon and their graphical representations offer critical insights into these processes. In exponential growth, such as population expansion, compound interest in finance or the spread of infectious diseases, the graph starts slowly but accelerates rapidly as time progresses. The curve increases steeply, showcasing how growth becomes more pronounced with time due to constant proportional changes. Conversely, in exponential decay, observed in cooling of objects or depreciation of assets, the graph starts high and decreases sharply before levelling off, indicating a gradual reduction over time. These graphs are essential for interpreting trends, making predictions and informing decision-making in diverse fields.

Example 12

The population of a village was $753$ in $2010$. If the population grows according to the equation $p = 753e^{0.03t}$, where $p$ is the number of persons in the population at time $t$,

(a) Graph the population equation for $t = 0$ (in $2010$) to $t = 30$ (in $2040$).
(b) From the graph, estimate the population (i) in $2020$ and (ii) in $2030$.

Solution: (a) The general shape of the exponential is known; however, since the graph is being used for estimations, an accurate graph over the required interval, $t = 0$ to $t = 30$, is required.

Calculate a table of values for different time periods and sketched in below figure:

[Graph of population growth over time]

(b) From graph,
(i) In $2020$ ($t = 10$) the population is $1016$ persons.
(ii) In $2030$ ($t = 20$) the population is $1372$ persons.