College

6-3: MathXL for School: Additional Practice Copy I

A wildlife biologist determines that there are approximately 200 deer in a region of a national park. The population grows at a rate of 7% per year. What is an exponential function that models the expected population?

A. [tex]f(x) = 200(0.07)^x[/tex]
B. [tex]f(x) = 200(1.07)^2[/tex]
C. [tex]f(x) = 1.07(200)^x[/tex]
D. None of the above

Answer :

To determine the expected population of deer in a region of a national park, we are given the following information:
- The initial population of deer is 200.
- The population grows at a rate of 7% per year.

From this information, we need to develop an exponential function that models the expected population over time. Here are the steps to solve this problem:

### Step-by-Step Solution:

1. Understanding the Growth Rate:
- A growth rate of 7% per year means that each year the population grows by 7% of its current size.
- An increase of 7% is the same as multiplying the population by 1.07 each year.

2. Exponential Growth Model:
- Exponential growth can be modeled by the function [tex]\( f(x) = a \cdot (b)^x \)[/tex], where:
- [tex]\( a \)[/tex] is the initial amount (starting population).
- [tex]\( b \)[/tex] is the growth factor (1 + growth rate in decimal form).
- [tex]\( x \)[/tex] is the number of time periods (years).

3. Determine the Components:
- The initial population ([tex]\( a \)[/tex]) is 200 deer.
- The growth factor ([tex]\( b \)[/tex]) is 1.07 (since 7% growth rate means each year's population is 107% of the previous year's population).

4. Construct the Function:
- Using the given values, the exponential function modeling the deer population [tex]\( f(x) \)[/tex] will be:
[tex]\[
f(x) = 200 \cdot (1.07)^x
\][/tex]

### Conclusion:

The exponential function that models the expected population of deer in the national park is:
[tex]\[ f(x) = 200 \cdot (1.07)^x \][/tex]

This function will provide the number of deer expected after [tex]\( x \)[/tex] years.