Question

(T) The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles.

(a) Use a power function to model N as a function of A.

(b) The Caribbean island of Dominica has area \(291\,\,{{\mathop{\rm mi}\nolimits} ^2}\). How many species of reptiles and amphibians would you expect to find on Dominica?

Short answer

a. The power regression equation is \(N = 3.1046{A^{0.308}}\).

b. You would expect to find 18 species of reptiles and amphibians on Dominica.

Step-by-step solution

Power function

A function of the form \(f\left( x \right) = {x^a}\) with \(a\) is a constant, is known as apower function.

Use a power function to model N as a function of A

a)

Consider the values of\(A\)as\(x\)-coordinates and values of\(N\)as\(y\)-coordinates.

The procedure to use the computing device to obtain the power regression equation as shown below:

1. Open the power regression calculator. Enter all the values of\(x\)in the\(x\)tab and\(y\)values in the\(y\)tab.
2. Select the “Calculate” button in the power regression equation calculator.

Obtain the power regression equation as shown below:

The power regression equation is\(y = 3.1046{x^{0.308}}\).

Use the power function to model N as the function of A is shown below:

\(N = c{A^b}\)

Where\(c \approx 3.1046\)and\(b \approx 0.308\).

Thus, the power regression equation is \(N = 3.1046{A^{0.308}}\).

Determine the species of reptiles and amphibians you expect

b)

When\(A = 291\)then

\(\begin{aligned}N &= c{A^b}\\ &= 3.1046{\left( {291} \right)^{0.308}}\\ &= 3.1046\left( {5.7395} \right)\\ \approx 17.819\end{aligned}\)

Thus, you would expect to find 18 species of reptiles and amphibians on Dominica.