Problem 82 - A Graphical Approach to College Algebra

Question

Solve each problem. A research biologist has shown that the number \(S\) of different plant species on a Galápagos Island is related to the area \(\mathscr{A}\) of the island by $$S=28.6 \sqrt[3]{\mathscr{A}}$$ Find \(S\) for an island with each area. (a) 100 square miles (b) 1500 square miles

Step-by-Step Solution

Answer

(a) S ≈ 132.70; (b) S ≈ 327.07

Step 1: Understand the Formula

The formula given is \(S = 28.6 \sqrt[3]{\mathscr{A}}\), where \(S\) is the number of plant species and \(\mathscr{A}\) is the area of the island. Our task is to plug in the values of \(\mathscr{A}\) and compute \(S\).

Step 2: Calculate S for 100 square miles

Substitute \(\mathscr{A} = 100\) into the formula.\[S = 28.6 \sqrt[3]{100} = 28.6 \times 100^{\frac{1}{3}}\]Find the cube root of 100, which is approximately 4.64.\[S = 28.6 \times 4.64 \approx 132.70\]Thus, the number of plant species for an island with an area of 100 square miles is approximately 132.70.

Step 3: Calculate S for 1500 square miles

Substitute \(\mathscr{A} = 1500\) into the formula.\[S = 28.6 \sqrt[3]{1500} = 28.6 \times 1500^{\frac{1}{3}}\]Find the cube root of 1500, which is approximately 11.45.\[S = 28.6 \times 11.45 \approx 327.07\]Thus, the number of plant species for an island with an area of 1500 square miles is approximately 327.07.

Key Concepts

Cube Root

The cube root is a mathematical operation which finds a number that, when multiplied by itself twice, gives the original number. For instance, the cube root of 8 is 2 because when 2 is multiplied by itself twice (2 \(\times\) 2 \(\times\) 2), the result is 8. In mathematical notation, the cube root is often represented as \(\sqrt[3]{x}\), where \(x\) is the number you need the cube root of.
If you have the number 100 as in the exercise, finding the cube root means finding a number that multiplies by itself twice to return 100. You can calculate 100's cube root as approximately 4.64 because 4.64 \(\times\) 4.64 \(\times\) 4.64 gives you a value near to 100.
Understanding cube roots is essential when dealing with formulas and functions, particularly when the formula involves an exponent of \(\frac{1}{3}\), indicating a hierarchical relationship or transformation such as growth or scale.

Function Evaluation

Function evaluation in mathematics involves calculating the output of a function once the input is known. For instance, if you are given a function \(f(x) = x^2\), and you need the outcome when \(x = 3\), then you substitute 3 with \(x\) and compute \(f(3) = 3^2 = 9\).
In the given exercise, the formula \(S = 28.6 \sqrt[3]{\mathscr{A}}\) is a function where \(S\) depends on \(\mathscr{A}\), the area's cube root, multiplied by 28.6. The task boils down to plugging in the given values of \(\mathscr{A}\) and finding the corresponding \(S\).
When \(\mathscr{A} = 100\), you compute \(S = 28.6 \times 100^{\frac{1}{3}} = 28.6 \times 4.64 \approx 132.70\). Similarly, with \(\mathscr{A} = 1500\), you get \(S = 28.6 \times 1500^{\frac{1}{3}} \approx 327.07\). This process demonstrates how essential function evaluation is in mathematical computations and scientific applications.

Scientific Research Application

Scientific research often employs mathematical models to predict behavior in nature, such as the growth or distribution of species. In the exercise, the biologist uses the formula \(S = 28.6 \sqrt[3]{\mathscr{A}}\) as a model to approximate the number of plant species based on the island's area.
This equation implies a relationship where the number of species scales with the cube root of the island's area, highlighting an ecological balance between area and biodiversity. Such relationships can support predictions and hypotheses in ecological research and conservation planning.
By assigning scientific understanding to mathematical constructs, researchers can draw insightful conclusions, make predictions, and offer recommendations. For instance, knowing that a larger island may support a greater number of species can guide measures to protect larger areas, maintain biodiversity, and inform land management strategies.
This blending of math and science showcases how algebraic principles translate into practical applications within the realm of scientific discovery and environmental studies.