The exponent $(2/5)t$ gives you the number of 7% growth periods that have occurred after $t$ years. How long (what value of $t$) makes this exponent equal to 1, meaning one 7% period?

Once you solve $(2/5)t = 1$ for $t$, you will have the time for one 7% increase in years. How do you convert that number of years into months to find $n$?

In Desmos, type the equation (2/5)x = 1. Desmos will show the value of x that makes this true; that value is the time in years for one 7% increase.

On a new Desmos line, type 12 * followed by the x-value you just found (for example, 12 * 2.5 if that was the solution). The output is the length of one growth period in months, which is the value of $n$.

The model is

Here:

• $480$ is the initial size (in thousands).
• $1.07$ is the growth factor for one compounding period (a 7% increase each period).
• The exponent $(2/5)t$ tells you how many 7% periods have happened after $t$ years.

So we need to find how long it takes (in time) for the exponent to increase by 1, because that corresponds to one 7% increase.

One 7% increase happens when the exponent $(2/5)t$ increases by 1.

So set the exponent equal to 1 and solve for $t$:

Multiply both sides by $5/2$:

So one 7% increase happens every $5/2$ years, which is 2.5 years.

We are told the colony increases by 7% every $n$ months, so we must convert the period from years to months.

There are 12 months in 1 year, so:

This product gives the value of $n$ in months, but we will simplify it in the next step.

Now multiply:

So the colony increases by 7% every $30$ months, which matches choice D.