The Galápagos Islands, lying 600 miles west of Ecuador, are...

The Galápagos Islands, lying 600 miles west of Ecuador, are famed for their extraordinary wildlife. The function $$f(x)=29 x^{\frac{1}{3}}$$ models the number of plant species, $f(x),$ on the various islands of the Galápagos chain in terms of the area, $x,$ in square miles, of a particular island. How many species of plants are on a Galápagos island that has an area of 27 square miles?

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Key Concepts

Function Evaluation

Function evaluation involves substituting a given input value into a function to determine the corresponding output. This process is fundamental in understanding how function rules operate over their domains and in using mathematical models to predict outcomes based on changes in input variables.

Fractional Exponents

Fractional exponents, such as an exponent of one-third, indicate roots. Specifically, an exponent of 1/3 represents the cube root of a number. Mastery of fractional exponents is key in simplifying and evaluating expressions that involve roots, enhancing the ability to solve a wide range of algebraic problems.

Mathematical Modeling

Mathematical modeling uses functions to represent real-world phenomena. By creating a function that describes the relationship between two quantities, such as island area and species count, it is possible to predict outcomes and draw insights from empirical data. This technique is crucial in fields like ecology, economics, and engineering.

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