Let's examine the problem one step at a time, using the given population model of deer in a forest:
[tex]\[ P(x) = \frac{2040x + 1820}{6x + 7} \][/tex]
where [tex]\( x \)[/tex] represents the number of years in the future.
1. How many deer live in this forest this year?
To determine the number of deer living in the forest this year, we evaluate [tex]\( P(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
P(0) = \frac{2040 \cdot 0 + 1820}{6 \cdot 0 + 7} = \frac{1820}{7} = 260.0
\][/tex]
Therefore, 260 deer live in the forest this year.
2. How many deer will live in this forest 16 years later?
To determine the number of deer living in the forest 16 years from now, we evaluate [tex]\( P(x) \)[/tex] at [tex]\( x = 16 \)[/tex]:
[tex]\[
P(16) = \frac{2040 \cdot 16 + 1820}{6 \cdot 16 + 7} = \frac{32640 + 1820}{96 + 7} = \frac{34460}{103} \approx 334.5699
\][/tex]
Rounding this to the nearest integer, the population 16 years later will be approximately 335 deer.
3. After how many years will the deer population be 330?
To find out after how many years the deer population will be 330, we need to set [tex]\( P(x) \)[/tex] equal to 330 and solve for [tex]\( x \)[/tex]:
[tex]\[
330 = \frac{2040x + 1820}{6x + 7}
\][/tex]
Solving this equation:
[tex]\[
330 (6x + 7) = 2040x + 1820
\][/tex]
[tex]\[
1980x + 2310 = 2040x + 1820
\][/tex]
[tex]\[
2310 - 1820 = 2040x - 1980x
\][/tex]
[tex]\[
490 = 60x
\][/tex]
[tex]\[
x = \frac{490}{60} \approx 8.1667
\][/tex]
Rounding to the nearest integer, the population will be 330 after approximately 8 years.
4. Many years in the future, about how many deer will live in this forest?
To find the asymptotic population of the deer as [tex]\( x \)[/tex] approaches infinity, we consider the dominant terms in the numerator and denominator of [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = \frac{2040x + 1820}{6x + 7} \approx \frac{2040x}{6x} = \frac{2040}{6} = 340
\][/tex]
Therefore, many years in the future, the deer population will approach approximately 340.
In summary:
1. Currently, there are 260 deer in the forest.
2. In 16 years, there will be approximately 335 deer.
3. The deer population will be 330 after approximately 8 years.
4. Many years in the future, the deer population will approach approximately 340.
