Math CentralQuandaries & Queries


Question from Paul, a parent:

The value of a stamp was 0.03 in 1851. The value in 2011 is $\$50,000.$ If the value increased exponentially, what is the annual growth rate?

Hi Paul,

I am going to let $t$ be the time in years where $t = 0$ in 1851. Since the value $V$ in dollars increases exponentially the value is

\[V = A e^{r \; t} \]

where $A$ and $r$ are constants. $r$ is called the exponential growth rate. Since $e^0 = 1$ at time $t = 0, V = A e^0 = A$ and hence $A$ is the value of the stamp at time $t = 0,$ that is in 1851. Hence

\[V = 0.03 e^{r \; t}.\]

What is $t$ in the year 2011? The value then is $\$50,000.$ Solve for $r.$

I hope this helps,

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