Chapter 4.5, Problem 37E

(a)

To determine

To graph: The function.

Explanation of Solution

Given information: Ecology: In the early 1900s, the deer population of the Kaibab Plateau in Arizona experienced a rapid increase because hunters had reduced the number of natural predators. The food supply was not great enough to support the increased population, eventually causing the population to decline. The deer population for the period 1905 to 1930 can be modeled by $f\left(x\right)=-0.125x^5+3.125x^4+4000,$ where x is the number of years from 1905.

Calculation:

The required graph is as follows:

  

(b)

To determine

To calculate: The population in 1905 using the model.

Answer to Problem 37E

4000 Deer

Explanation of Solution

Given information: Ecology: In the early 1900s, the deer population of the Kaibab Plateau in Arizona experienced a rapid increase because hunters had reduced the number of natural predators. The food supply was not great enough to support the increased population, eventually causing the population to decline. The deer population for the period 1905 to 1930 can be modeled by $f\left(x\right)=-0.125x^5+3.125x^4+4000,$ where x is the number of years from 1905.

Calculation:

1905 coincides with $x=0,$

  $\begin{array}{l}f\left(0\right)=-0.125{\left(0\right)}^5+3.125{\left(0\right)}^4+4000=4000\\ \Rightarrow 4000\text{ Deer}\end{array}$

Thus, population is 4000 deer.

(c)

To determine

To find: The population in 1920 using the model.

Answer to Problem 37E

67,281 Deer

Explanation of Solution

Given information: Ecology: In the early 1900s, the deer population of the Kaibab Plateau in Arizona experienced a rapid increase because hunters had reduced the number of natural predators. The food supply was not great enough to support the increased population, eventually causing the population to decline. The deer population for the period 1905 to 1930 can be modeled by $f\left(x\right)=-0.125x^5+3.125x^4+4000,$ where x is the number of years from 1905.

Calculation:

1920 coincides with 15 years after 1905.

  $f\left(15\right)=-0.125{\left(15\right)}^5+3.125{\left(15\right)}^4+4000=67,281.25$

This would estimate about 67,281 deer.

(d)

To determine

To find: When did the deer population become zero, according to this model.

Answer to Problem 37E

1930

Explanation of Solution

Given information: Ecology: In the early 1900s, the deer population of the Kaibab Plateau in Arizona experienced a rapid increase because hunters had reduced the number of natural predators. The food supply was not great enough to support the increased population, eventually causing the population to decline. The deer population for the period 1905 to 1930 can be modeled by $f\left(x\right)=-0.125x^5+3.125x^4+4000,$ where x is the number of years from 1905.

Calculation:

  $f\left(x\right)=0$ when x = 25 which means 1905 + 25 = 1930

  $\Rightarrow 1930$