Contextual problems with exponentials III

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

Question 2

Question 3