Problem 61 - A Graphical Approach to College Algebra

Question

Solve each problem. Wing Size Suppose that the surface area \(S\) of a bird's wings, in square feet, can be modeled by $$ S(w)=1.27 w^{2 / 3} $$ where \(w\) is the weight of the bird in pounds. Estimate the surface area of a bird's wings if the bird weighs 4.0 pounds.

Step-by-Step Solution

Answer

The surface area of the bird's wings is approximately 3.20 square feet.

Step 1: Understand the Formula

The formula given is \( S(w) = 1.27 w^{2/3} \), which estimates the surface area \( S \) of a bird's wings based on its weight \( w \). The \( w^{2/3} \) part is an important calculation involving exponents.

Step 2: Substitute Weight into Formula

We are given that the bird weighs 4.0 pounds. Substitute \( w = 4.0 \) into the formula:\[ S(4.0) = 1.27 (4.0)^{2/3} \]

Step 3: Calculate the Exponent

Calculate \( (4.0)^{2/3} \). First, take the cube root of 4.0, which is \( 4.0^{1/3} = \sqrt[3]{4} \approx 1.5874 \), then square the result to get \( 1.5874^2 \approx 2.5198 \).

Step 4: Complete the Surface Area Calculation

Substitute \( 2.5198 \) back into the equation: \[ S(4.0) = 1.27 \times 2.5198 \] which equals \( S(4.0) \approx 3.2001 \) square feet.

Key Concepts

Surface Area

The surface area of a bird's wings is an essential metric in ornithology and physics because it has significant implications for the bird's flight capabilities. Here, we are modeled to estimate the surface area using the formula:

• \( S(w) = 1.27 w^{2/3} \)
• Where \( S \) is the surface area and \( w \) is the weight of the bird in pounds.

Understanding how to manipulate this formula is crucial. In this scenario, the surface area will change based on the weight through the exponent applied. The formula shows that the surface area does not increase linearly with weight but instead follows a power law, represented by the exponent \( 2/3 \). This relationship indicates that heavier birds do not require proportionately larger wing areas.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena through mathematical formulas and equations. In the case of the bird's wing exercise, the model \( S(w) = 1.27 w^{2/3} \) is used to estimate wing surface area based on weight.

• These models help predict outcomes based on variable inputs.
• They simplify complex biological systems into manageable equations.

In this problem, the exponent of \( 2/3 \) suggests a relationship derived from empirical data, experiments, or theoretical analysis. The model provides not only a method for estimation but also insights into how animals are sized and how their physical traits evolve to optimize their roles in their respective ecosystems. Models like these allow ecologists and scientists to make predictions about animal behavior, create conservation strategies, and understand animal adaptability.

Weight

Weight is a fundamental concept in biology and physics, often influencing an organism's physical capabilities. When examining the effect of weight on wing surface area, it's important to consider how it interacts with formulas.

• Weight, measured here in pounds, is a critical input in our given formula.
• It directly affects the outcome of the surface area calculation as it undergoes a transformation through exponentiation.

With the exponent of \( 2/3 \), the model indicates that as a bird increases in weight, the corresponding increase in wing surface area is not proportional. The exponentiation process highlights an intricate relationship where the square root and cube root operations are involved, reflecting the complexities of biological scaling. Understanding this context helps us appreciate how factors like weight influence the physical design of an organism and contribute to their survival and efficiency in flight. This adaptation is crucial, allowing birds of different weights to maintain efficient flight capabilities.