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 and, in particular, the number of species of bats living in caves in central Mexico has been related to the surface area\({\bf{A}}\)of the caves by the equation\(S = 0.7{A^{0.3}}\).

(a) The cave called MisionImposible near Puebla, Mexico, has a surface area of\({\bf{A = 60}}\;{{\bf{m}}^{\bf{2}}}\). 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) There are\(2\;species\)of bats are in the cave.

(b) The surface area would be around \(334\;{m^2}\)

Step-by-step solution

Given data

The number of species of bats \(S\) is dependent on the surface area \(A\)of a cave and is given by\(S = 0.7{A^{0.3}}\).

Step 2:(a) Find how many species of bats are in the cave

The surface area is \(60\) so we plug in and find the value of\(S\), namely

\begin{aligned}& S=0.7\left( {{60}^{^{0.3}}} \right) \\ & =0.7{{\left( 60 \right)}^{\frac{3}{10}}} \\ & =0.7\sqrt(10){{{60}^{3}}}\approx 2.4 \\& \cong 2 species \end{aligned}

Hence, \(2\;species\)of bats are in the cave.\(\)

(b) Estimate the area of the cave

The value of\(S\)is\(4\)so we plug in and find the value of\(A\), namely

\(\)\(\begin{array}{c}{A^{0.3}} = \frac{4}{{0.7}}\\{\left( {{A^{0.3}}} \right)^{^{\frac{1}{{0.3}}}}} = {\left( {\frac{4}{{0.7}}} \right)^{\frac{1}{{0.3}}}}\\A = {\left( {\frac{{40}}{7}} \right)^{\frac{1}{{0.3}}}}\\A = 333.6\;{m^2}\end{array}\)(Take the power \(\frac{1}{{0.3}}\) on both sides)

Hence, the surface area would be around\(334\;{m^2}\).