Question

Suppose that the future price of a certain item is given by the following exponential function. In this function, is measured in dollars and is the number of years from today.

Answer 1

Solution

- The general formula for the exponential growth or decay is given below:

$\begin{gathered} P(t)=P_0(1+r)^t \\ where, \\ P_0=\text{ Initial amount} \\ r=\text{ The rate of decay or growth. } \\ \text{ If }r\text{ is positive, then, it is a growth while if }r\text{ is negative, it is a decay} \\ t=\text{ The time elapsed.} \end{gathered}$

- Comparing this formula with the equation given to us, we have:

$\begin{gathered} P(t)=1800(1.022)^t \\ P(t)=P_0(1+r)^t \\ \\ P_0=1800.\text{ Thus, the initial price is 1800} \\ \\ 1+r=1.022 \\ Subtract\text{ 1 from both sides} \\ r=1.022-1 \\ r=0.022=\frac{2.2}{100}=2.2\% \\ \\ \text{ Since r is positive, then, this represents a GROWTH} \end{gathered}$

Final Answer

- The initial price is 1800

- This is a GROWTH

- The rate of growth is 2.2%