# The function P= 1300(1.7)d models the number of deer in a region after d decades. What...

## Question:

The function P= 1300(1.7)d models the number of deer in a region after d decades. What exponential function models the number of deer after t years? What is the annual growth rate of the number of deer?

## Exponential Function

Since an exponential function contains a variable inside of an exponent, we have an infinite amount of possible exponential functions corresponding to an infinite amount of numerical bases. We can convert exponential functions from one form to another by making sure they agree at two points, as we need to define both the leading coefficient and any constant that lies next to the variable for a given base.

Since this function is given in terms of decades, we can convert it to a function that uses years as an input instead. First, let's write an expression that converts decades to years:

{eq}d = 0.1t {/eq}

Let's substitute this into our function to construct a new function that gives us the population of deer after a number of years rather than a number of decades.

{eq}P = 1300(1.7)^{0.1t} {/eq}

We can now find the annual growth rate of this population. This is because the growth rate should be constant, as it is for every exponential function. We can evaluate this function at two consecutive years and then take the ratio of the two numbers we find. This will give us our growth rate.

{eq}P(0) = 1300(1.7)^{0.1(0)} = 1300\\ P(1) = 1300(1.7)^{0.1(1)} = 1370.84466\\ \displaystyle \frac{1370.84466}{1300} \approx 1.054496 {/eq}

Thus, each year, the population is 105.45% of the previous year's population, making the annual growth rate 5.45%.