Chapter 4, Problem 7P

Modeling the Species-Area Relation The table gives the areas of several caves in central Mexico and the number of bat species that live in each cave.*

1. (a) Find a power function that models the data.
2. (b) Draw a graph of the function you found in part (a) and a scatter plot of the data on the same graph. Does the model fit the data well?
3. (c) The cave called El Sapo near Puebla, Mexico, has a surface area of A = 205 m². Use the model to estimate the number of bat species you would expect to find in that cave.

Cave	Area (m2)	Number of species
La Escondida	18	1
El Escorpion	19	1
El Tigre	58	1
Mision Imposible	60	2
San Martin	128	5
El Arenal	187	4
La Ciudad	344	6
Virgen	511	7

To determine

To find: The power function that models the data.

Answer to Problem 7P

The power function that gives the best fits of the data point is $y=0.14\times x^{0.64}$ , where the variables $y$ represents the number of species in cave and $x$ represents the area of the cave.

Explanation of Solution

Given:

The data is in the tabular form as shown below.

$\mathrm{Cave}$	$\mathrm{Area}\left(m^2\right)$	$\mathrm{Number}ofspecies$
La Escondida	18	1
El Escorpion	19	1
El Tiger	58	1
Mision Imposible	60	2
San Martin	128	5
El Arenal	187	4
La Ciudad	344	6
Virgen	511	7

Use the graphing calculator, the power function of the form $y=a{t}^b$ that gives the best fit of the data point is $y=0.14\times x^{0.64}$, where the variables $y$ represents the number of species in cave and $x$ represents the area of the cave.

Therefore, the power function that gives the best fit of the data point is $y=0.14\times x^{0.64}$ , where the variables $y$ represents the number of species in cave and $x$ represents the area of the cave.

To determine

To draw: The graph of the power function is $y=0.14\times x^{0.64}$ and scatter the plot of the data on the same graph and then explain does the model fit the data well.

Answer to Problem 7P

The power function $y=0.14\times x^{0.64}$ fits the data well except one data point.

Explanation of Solution

Use online graphing calculator and draw the scatter plot of the given data and the graph of the power function $y=0.14\times x^{0.64}$ on the scatter plot as shown below in Figure 1.

In Figure 1, the grey circle represents the scatter plot and the black line represents the power function.

From Figure 1, it is noticed that the power function $y=0.14\times x^{0.64}$ fits the data sufficiently well except one point which is moving away from the power function.

Therefore, the power function $y=0.14\times x^{0.64}$ fits the data well except one data point.

To determine

To estimate: The number of bat species in the cave called EL Sapo near Puebla, has a surface area of $A=205{\mathrm{m}}^2$ by using the power model.

Answer to Problem 7P

The number of bat species in the cave called EL Sapo near Puebla of the surface area of $A=205{\mathrm{m}}^2$ is $y=4.22$.

Explanation of Solution

From part (a), the power function that gives the best fits of the data point is $y=0.14\times x^{0.64}$ , where the variables $y$ represents the number of species in cave and $x$ represents the area of the cave.

Substitute, $x=205$ in $y$, then the number of species in the cave EL Sapo near Puebla is calculated as follows,

$\begin{array}{c}y=0.14\times {\left(205\right)}^{0.64}\\ =4.22\end{array}$

Therefore, the number of bat species in the cave called EL Sapo near Puebla of the surface area of $A=205{\mathrm{m}}^2$ is $y=4.22$.