Problem 71 - A Graphical Approach to College Algebra

Question

Wing Size (See Exercise 61.) Heavier birds have larger wings with more surface area than do lighter birds. For some species of birds, this relationship is given by $$ S(x)=0.2 x^{2 / 3} $$ where \(x\) is the weight of the bird in kilograms and \(S\) is the surface area of the wings in square meters. (Source: Pennycuick, C., Newton Rules Biology, Oxford University Press.) Approximate \(S(0.5)\) and interpret the result.

Step-by-Step Solution

Answer

For a 0.5 kg bird, the wing surface area is approximately 0.126 square meters.

Step 1: Identify the Given Function

The given function is \( S(x) = 0.2 x^{2/3} \). This function describes the relationship between the bird's weight \( x \) in kilograms and the wing surface area \( S \) in square meters.

Step 2: Substitute the Given Value

We are asked to approximate \( S(0.5) \). Substitute \( x = 0.5 \) into the function: \( S(0.5) = 0.2 (0.5)^{2/3} \).

Step 3: Calculate the Exponent

Calculate \( (0.5)^{2/3} \). To do this, first find the cube root of 0.5 and then square the result. \[ (0.5)^{2/3} = (0.5^{1/3})^2 = (0.7937)^2 \approx 0.6299 \].

Step 4: Multiply by 0.2

Multiply the result from Step 3 by 0.2 to find \( S(0.5) \). \[ S(0.5) = 0.2 \times 0.6299 \approx 0.12598 \text{ square meters} \].

Step 5: Interpret the Result

The approximated surface area of the wings for a bird weighing 0.5 kilograms is about 0.126 square meters. This means that a lighter bird with this weight has a wing surface area proportionate to its weight according to the given relationship.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation where a number, called the base, is raised to a power, known as the exponent. This concept occurs frequently in algebra and is defined as the repeated multiplication of the base. For instance, in the expression \( a^n \), \( a \) is the base and \( n \) is the exponent, indicating that \( a \) should be multiplied by itself \( n \) times.
In the context of our bird wing size function \( S(x) = 0.2 x^{2/3} \), exponentiation is used with a fractional exponent. The exponent \( \frac{2}{3} \) signifies that we must:

• Calculate the cube root of \( x \) (because of \( 1/3 \)).
• Then, square that result (because of \( 2 \)).

Working with fractional exponents like \( \frac{2}{3} \) may initially appear daunting, but it is simply a combination of root and power operations.
For example, when calculating \( (0.5)^{2/3} \), you'll first find the cube root of 0.5, which is approximately 0.7937. Then, you square this result, obtaining approximately 0.6299. Using exponentiation in such a formula helps model real-world phenomena in an elegant and mathematically sound way.

Function Evaluation

Function evaluation involves substituting a specific value into a function to calculate an output. This process is essential when you want to predict or determine specific inputs' results within a mathematical model.
For the function \( S(x) = 0.2 x^{2/3} \), to evaluate \( S(0.5) \), you replace \( x \) with 0.5 in the equation. This substitution results in the expression \( S(0.5) = 0.2 (0.5)^{2/3} \).
Next, solve the exponentiation \( (0.5)^{2/3} \), as explained in the previous section. Once you have that figure, you multiply it by 0.2 to complete the function evaluation. Function evaluation is a powerful tool that provides insights into how variables interact within mathematical models and helps to understand the relationship between different quantities.
By evaluating \( S(0.5) \), we derive the wing surface area for a bird weighing 0.5 kilograms, allowing us to interpret how smaller weights influence the surface area of bird wings, thus reflecting the direct real-life application of this mathematical principle.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world situations. It is a critical tool in fields such as biology, engineering, economics, and more, where it helps to predict behaviors and understand complex systems.
In our exercise, the function \( S(x) = 0.2 x^{2/3} \) models the relationship between a bird's weight and its wing surface area. This model stems from observations that heavier birds typically have more extensive wings to support their weight.
The equation demonstrates a proportional relationship with a specific rate determined through empirical research. By inputting a weight, such as 0.5 kilograms, into this function, we ascertain the approximate wing surface area—0.126 square meters in this case.
Mathematical models, like the one used here, are beneficial because they simplify complex biological relationships into precise and predictable outcomes. Additionally, they provide scientists and scholars with tools for predicting phenomena based on quantifiable metrics, thereby enhancing our understanding of the natural world.