Applications of Exponential Functions (1 transform, positive base)
Two quantities have an exponential relationship if whenever one quantity changes by a unit, the other is multiplied by a constant factor. Consequently, this simple relationship describes many situations such as the value of a term deposit into a savings account or the growth of a population.
Exploration
EXAMPLE 1
The value of a term deposit $P$P in dollars at time $t$t given in years can be given by the equation:
$P=A\left(1+r\right)^t$P=A(1+r)t
where $A$A is the initial term deposit and $r$r is the annual interest rate. As we can see, $P$P grows exponentially since for every year that passes, we multiply the previous term deposit by a factor of $1+r$1+r.
EXAMPLE 2
Exponential relationships also arise in biology. We might wish to count the total number of bacteria cells that have been cultivated on a petri dish. If there is initially $1$1 live cell and $2$2 dead cells, how can we find the total number of cells after $t$t days given that the live cells double every day? The total number of cells $T$T after $t$t days can be given by the equation:
$T=2^t+2$T=2t+2
where the number of live cells is $2^t$2t and the number of dead cells remains constant at a value of $2$2.
EXAMPLE 3
A model of population growth beginning with only ten animals of a particular species uses the observed fact that the number of individuals doubles every month. Thus, if there are $n$n individuals at a given time, there will be $2n$2n in another month. If $t$t denotes the amount of time that has passed in months, then we can construct an equation relating $n$n and $t$t as follows:
$n=10\left(2^t\right)$n=10(2t)
For every month that passes, we multiply the previous value of $n$n by two. Of course when $t=0$t=0, we get the initial number of animals, that being $n=10$n=10.
Practice questions
Question 1
The number of fungal cells, $N$N, in a colony after $t$t hours is given by the equation $N=5000\left(4^t\right)$N=5000(4t).
Determine the initial population of fungal cells.
Determine the population of fungal cells after $8$8 hours.
Plot the graph of fungal cell population over time.
According to the graph, approximately how many hours will it take for the population to reach $3$3 times the original population?
Question 2
Peter received a lump sum payment of $£50000$£50000 for an insurance claim (and decided not to put it in a savings account).
Every month, he withdraws $5%$5% of the remaining funds.
The funds after $x$x months is shown below.
How much would the first withdrawal be?
How much would be left after the first withdrawal?
According to the graph, which of the following is the best estimate for the amount left after one year?
$£0$£0
A$£3000$£3000
B$£27000$£27000
C$£30000$£30000
DAccording to the graph, which of the following is the best estimate for the number of months until the amount reaches $£30000$£30000?
$5$5
A$10$10
B$20$20
C$40$40
D
Question 3
A new appliance is valued at $£990$£990.
Each year it is worth $9%$9% less than the previous year's value.
Calculate the value of the appliance after the first year to the nearest penny.
Calculate the value of the appliance after the second year to the nearest penny.
Determine the equation that relates the value of the appliance, $A$A, with the number of years passed, $t$t.
Using the equation in part (c), calculate the value of the appliance after twelve years to the nearest penny.