Chapter 4.4, Problem 62PS

Solution

i.

To graph: The function using graphing calculator.

Energy magnitude $=6.4$

Given:

The function $s=30.6-20.5\left(\log A\right)+3.8{\left(\log A\right)}^2$.

Explanation:

The following graph has been obtained using the graphing calculator.

  

ii.

To calculate: The number of fish species within the specified area.

  $15$

Given:

The function $s=30.6-20.5\left(\log A\right)+3.8{\left(\log A\right)}^2$ and area is $30,000{\text{ m}}^2$.

Calculation:

The $x$ -axis of the graph is representing the area and the $y$ -axis is representing the species. So, when $x=30000$ the value of $y$ is $14.989$ that can be observed in the following graph.

  

Conclusion:

The number of fish species is $15$.

iii.

To calculate: The area that contains the $6$ species of fish.

  $3917{\text{ m}}^2$

Given:

The function $s=30.6-20.5\left(\log A\right)+3.8{\left(\log A\right)}^2$ and $6$ species of fish.

Calculation:

The number of species is $6$ . It means $y=6$ . So,

  

From the graph, the estimated area is $3917{\text{ m}}^2$.

Conclusion:

The estimated area is $3917{\text{ m}}^2$.

iv.

To explain: The effect of area on the number of species.

On increasing area, the number of species will increase.

Given:

The function $s=30.6-20.5\left(\log A\right)+3.8{\left(\log A\right)}^2$.

Calculation:

The graph of function is as follows:

  

The curve of the graph rises as we move towards the infinity. So, it can be concluded that the number of species will also rise because $y$ -axis representing species here.

Conclusion:

The number of species will increase on increasing area.