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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 $ S $ of bats living in caves in central Mexico has been related to the surface area $ A $ of the caves by the equation $ S = 0.7A^ {0.3} $.

(a) The cave called Mision Imposible near Puebla, Mexico, has a surface area of $ A = 60 m^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.

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01:43

Jeffrey Payo

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 2

Mathematical Models: A Catalog of Essential Functions

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Johns Hopkins University

Harvey Mudd College

Idaho State University

Boston College

Lectures

04:31

A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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It makes sense that the la…

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It’s logical that the larg…

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Refer to a colony of bats …

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in this problem. We have a relationship between the number of species of bats on the surface area of the cave and for party were given a surface area of 60 square meters and we want to find the number of of species of bats there. So we're going to substitute 60 into the equation for a S equals 0.7 times 60 raised to the 0.3 power. And then we'll put that into a calculator and we get approximately 2.39 species and then in part B suppose there are four species in a cave. What's the surface area? So we're going to substitute for into the equation for s and solve for a We'll start by dividing both sides by 0.7. Now, I'm not going around anything just yet. I'm going to leave everything exact, and the next thing I'm going to do is raise both sides of the equation Toothy power of 1/0 0.3 that will clear out the 0.3 power on the other side. Now, if we put this into the calculator and keep everything exact and don't round until the very end, we get an area of approximately 333 0.58 square meters. However, if you instead round your four divided by 40.7 before you raised to the power that was given in the problem, you are going to get approximately 332 0.75 square meters. So rounding first before you do more does decrease your accuracy, but for both of these answers were in the ballpark of about 333 square meters.

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