Question

How do you write an exponential function to model the situation. Then predict the value of the function after 5 years (to the nearest whole number). A population of 430 animals that decreases at an annual rate of $12\%$?

Answer 1

Here we need to find the value of the exponential function for the given conditions. We will first use the definition of the exponential function and the expression we use for the exponential function. We will substitute all the given values in the expression. Then we will use this expression of the exponential function to find the number of animals after 5 years.

Complete step by step solution:
Here we need to find the value of the exponential function for the given conditions.
It is given that the rate of decrease of the population is $12\%$. We can write it as $1 - \dfrac{{12}}{{100}} = \dfrac{{88}}{{100}} = 0.88$
We know that the exponential function is of the form
$y = A{\left( r \right)^x}$ ………….. $\left( 1 \right)$
Where, $A$ is the initial value and $r$ is the rate of increase or decrease in decimals.
As we know there were 430 animals at the initial stage. So $A=430$.
As we also know that the rate of decrease in the population is equal to 0.88. So we can say that $r=0.88$.
We have to calculate the value of the function after 5 years. Therefore, $x = 5$
Now, we will substitute all the values here.
$\Rightarrow y = 430 \times {\left( {0.88} \right)^5}$
Now, we will apply the exponent on the base here.
$\Rightarrow y=430\times 0.5277$
Now, we will multiply the numbers here.
$\Rightarrow y=226.924\approx 227$

Hence, after 5 years, there will be 227 animals left.

Note:
Here we have used the exponential function and also we have also used the general equation for the exponential function. When the exponent in the exponential function is increased by the number 1, the value of the function increases by a factor of $e$. Similarly, when the exponent decreases by the number 1, then the value of the function decreases by this same factor. The logarithm function is the inverse of an exponential function. The exponential function is used to find the exponential growth or decay or to compute the investment graph.