Problem 71 - A Graphical Approach to Precalculus with Limits

Question

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

The surface area for a 0.5 kg bird is approximately 0.126 square meters.

Step 1: Understand the Function

The function given is \( S(x) = 0.2 x^{2/3} \). This function tells us how the surface area \( S \) of a bird's wings changes with its weight \( x \). The bird's weight \( x \) is in kilograms and the surface area \( S \) is in square meters.

Step 2: Substitute the Given Value

We need to find \( S(0.5) \). To do that, substitute \( x = 0.5 \) into the function: \( S(0.5) = 0.2 \, (0.5)^{2/3} \).

Step 3: Calculate the Exponent

Compute \( (0.5)^{2/3} \). First, find \( 0.5^{2} = 0.25 \). Then, take the cube root of \( 0.25 \), which is approximately \( 0.62996 \).

Step 4: Multiply by the Coefficient

Multiply the result by 0.2: \( S(0.5) = 0.2 \times 0.62996 = 0.125992 \).

Step 5: Interpret the Result

The approximate value of \( S(0.5) \) is 0.126 square meters. This means that a bird weighing 0.5 kilograms has wing surface area approximately equal to 0.126 square meters.

Key Concepts

Exponential Functions

Exponential functions often describe how quantities scale with size or time. In the exercise, the function \(S(x) = 0.2 x^{2/3}\) is an example of an exponential function that models the surface area of a bird's wings relative to its weight, \(x\).
This particular function uses an exponent of \(\frac{2}{3}\), indicating that the relationship between the weight of the bird and the wing's surface area is not linear.

• The base, \(0.5\), represents the bird's weight in kilograms.
• The exponent, \(\frac{2}{3}\), signifies a non-linear growth rate, meaning as the bird becomes heavier, its wing's surface area increases at a decreasing rate.
• The coefficient, \(0.2\), scales the surface area based on this relationship, ensuring that the model accurately reflects real bird data.

Understanding exponential functions is fundamental in precalculus, as they appear in various scientific models to explain growth patterns, decay, or scaling phenomena.

Surface Area Calculations

Surface area calculations are crucial when analyzing how organisms such as birds interact with their environment. In our exercise, the main focus is on calculating the surface area of a bird's wings given by the function \(S(x) = 0.2 x^{2/3}\).

• The given weight \(x\), when substituted into the function, allows you to compute the exact wing surface area.
• For example, substituting \(x = 0.5\) gives \(S(0.5) = 0.2 \times (0.5)^{2/3}\), which after calculation results in approximately \(0.126\) square meters.
• Calculating these areas helps researchers understand how changes in an organism's weight impact their abilities for flight and maneuverability.

Remember, surface area isn't just a theoretical concept—it's a practical measure that influences how creatures live and move.

Mathematical Interpretation

Mathematical interpretation bridges the gap between numbers and real-world meaning. Here, interpreting the solution \(S(0.5) \approx 0.126\) involves understanding what this translates to in terms of the bird's physical characteristics.
Interpreting this result, we see that a bird weighing 0.5 kilograms is predicted to have a wing surface area of 0.126 square meters. This result provides insight into how weight can influence flight capacity, with larger wings typically allowing for more efficient flight.

• A key aspect of interpretation is contextualizing results within biological and ecological frameworks—recognizing, for example, how surface area affects lift and drag in flight.
• Such interpretations allow scientists to make broader predictions about bird species, like migratory patterns or adaptations to different environments.

Developing strong interpretation skills is vital as it allows one to not only solve mathematical problems but to also understand their implications in real-world contexts.