Question

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function. In particular, the number of species S to bats living in caves in central Mexico has been related to the surface area A of the caves by the equation \(S = 0.7{A^{0.3}}\).

(a) The cave called Mission Impossible near Puebla, Mexico, has a surface area of \(A = 60{{\mathop{\rm m}\nolimits} ^2}\). How many species of bats would you expect to find in that cave?

(b) If you discover that four species of bats live in a cave, estimate the area of the cave?

Short answer

a. Two bat species are expected to be found in that cave.

b. The surface area of the cave is calculated to be \(334{{\mathop{\rm m}\nolimits} ^2}\).

Step-by-step solution

Determine the number of species of bats that you expect

a)

The surface area A of the caves as per the equation is given by \(S = 0.7{A^{0.3}}\).

Substitute \(A = 60\) in the above equation as shown below:

\(\begin{aligned}S = 0.7{\left( {60} \right)^{0.3}}\\ = 0.7\left( {3.4154} \right)\\ \approx 2.3908\end{aligned}\)

Thus, two bat species are expected to be found in that cave.

Estimate the area of the cave

b)

It is given that you discovered four species of bats live in a cave.

Substitute\(S = 4\)in the surface area of the caves equation as shown below:

\(4 = 0.7{A^{0.3}}\)

\(\frac{{40}}{7} = {A^{\frac{3}{{10}}}}\)

\(A = {\left( {\frac{{40}}{7}} \right)^{\frac{{10}}{3}}}\)

\( \approx 333.6\)

Thus, the surface area of the cave is calculated to be \(334{{\mathop{\rm m}\nolimits} ^2}\).