Problem 61 - A Graphical Approach to Precalculus with Limits

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 bind weighs 4.0 pounds.

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The surface area of a bird's wings is given by the function \( S(w) = 1.27 w^{2/3} \). Here, \( S \) represents the surface area in square feet, and \( w \) is the weight of the bird in pounds.

Step 2: Substitute the Weight

The problem asks for the surface area when the bird weighs 4.0 pounds. Substitute \( w = 4.0 \) into the function: \( S(4.0) = 1.27 (4.0)^{2/3} \).

Step 3: Calculate the Exponent

Calculate \( (4.0)^{2/3} \). First, find the cube root of 4, which is approximately 1.5874, and then square it: \( 1.5874^2 \approx 2.5208 \).

Step 4: Complete the Calculation

Multiply the result from Step 3 by 1.27 to find the surface area: \( S(4.0) = 1.27 \times 2.5208 \approx 3.20 \).

Step 5: Verify the Units

Ensure the resulting surface area is in square feet, which is consistent with the function's description.

Key Concepts

Exponentiation

Understanding exponentiation is crucial when dealing with mathematical functions, especially in applications like the one for modeling the surface area of a bird's wings. In our function, the exponent is expressed as a fraction \( \frac{2}{3} \). This suggests a two-step process: first, take the cube root of the weight \( w \), and then square the result.

This fractional exponent lets us work with real numbers that may not be whole.

• **First, the cube root is calculated.** For example, the cube root of 4 is around 1.5874 because \( 1.5874^3 \approx 4 \).
• **Next, square the cube root.** Continuing the example, squaring 1.5874 gives approximately 2.5208. This completes the calculation of \( 4^{2/3} \).

Exponents allow us to compactly express repeated multiplication and operate effectively on weights, volumes, and other properties in natural phenomena.

Surface Area Calculation

Surface area is an important concept in the study of physics, biology, and engineering, offering insights into various phenomena. It describes the total area covered by the surface of an object. In our context, we use it to model the area of a bird's wings with respect to its weight.

The formula for wing surface area is a specific application of a general model:\[ S(w) = 1.27 w^{2/3} \]Here, 1.27 is a constant that represents a proportionality factor in the model. It helps adjust the output of the function to more effectively fit real-world observations or measurements.

• Calculate the weight raised to a power as described with exponentiation.
• Multiply this result by the constant factor 1.27 to get the surface area.

This simple equation allows researchers and enthusiasts to predict a bird's wing surface area just by knowing its weight, giving vital insights into how birds fly.

Weight Conversion

Although this problem focuses on the weight in pounds, understanding weight conversion is a worthy exercise to broaden your understanding of mathematical modeling. Often, weights may be given in different units, such as grams or kilograms. Here’s how you can convert them to pounds if needed:

• **From grams to pounds:** Use the conversion factor \( 1 \text{ lb} = 453.592 \text{ g} \).
• **From kilograms to pounds:** There are 2.20462 pounds in a kilogram.

For example, if a bird's weight is given as 1800 grams, convert it to pounds by dividing by 453.592. Similarly, if it weighs 2.5 kilograms, multiply by 2.20462 to find the equivalent weight in pounds. These conversions are essential for accurate calculations in many scientific and mathematical contexts, ensuring compatibility with established models or functions. When dealing with different units, always remember to ensure consistency to maintain precision in your calculations.