Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Suppose that the wingspan \(L\) in feet of a bird weighing \(W\) pounds is given by $$L=2.43 W^{0.3326}$$ Estimate the wingspan of a bird that weighs 5.2 pounds.

Step-by-Step Solution

Answer

The wingspan is approximately 4.04 feet.

Step 1: Understand the Formula

The given formula to calculate the wingspan of a bird based on its weight is \( L = 2.43 W^{0.3326} \). Here, \( L \) represents the wingspan in feet and \( W \) represents the weight in pounds.

Step 2: Substitute the Given Weight

To find the wingspan for a bird weighing 5.2 pounds, substitute \( W = 5.2 \) into the formula. This gives \( L = 2.43 (5.2)^{0.3326} \).

Step 3: Calculate the Exponent

First, calculate \( 5.2^{0.3326} \). Use a calculator to find this value: \( 5.2^{0.3326} \approx 1.660 \).

Step 4: Multiply by the Constant

Now, multiply the result of the exponent by 2.43. So, calculate \( L = 2.43 \times 1.660 \approx 4.036 \).

Step 5: Interpret the Result

The wingspan of a bird weighing 5.2 pounds is approximately 4.04 feet.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation, fundamental in models like those estimating bird wingspans. It involves raising a number or variable to a power. In our exercise, we utilize the formula \( L = 2.43 W^{0.3326} \). Here, the weight \( W \) of the bird is raised to the power of 0.3326.

When we substitute a specific weight, say 5.2 pounds, into this model, we need to compute \( 5.2^{0.3326} \). This raw calculation tells us how the weight influences the wingspan when adjusted by a specific exponent like 0.3326.

Understanding this operation requires noting that exponents less than 1 yield results between 1 and the base number. So, \( 5.2^{0.3326} \) shrinks the impact of 5.2, compressing its influence on the outcome. Use a calculator to simplify such expressions, which, in our scenario, approximately results in 1.660.

Substitution

In mathematics, substitution is a method of introducing known values into an equation to find an unknown variable. In our wingspan problem, we substitute the weight of the bird into the formula to determine the wingspan.

To perform substitution accurately, ensure that you clearly identify the variable to replace. For example, the weight variable \( W \) in our formula is replaced by 5.2 pounds. This results in the modified expression: \( L = 2.43 (5.2)^{0.3326} \).

After substitution, the equation becomes entirely numeric, making it ready for calculation. Substitution transforms the theoretical formula into a practical tool, calculated step by step for definitive outcomes. This technique is crucial in many fields, offering a straightforward approach to applying theoretical concepts in real-world scenarios.

Formula Interpretation

Understanding how to interpret formulas is essential in recognizing the relationships between different variables. In our case, \( L = 2.43 W^{0.3326} \) gives a functional insight into how a bird's weight affects its wingspan.

The constant 2.43 scales the result of the exponential math, serving as a "multiplier" that adjusts the generalized findings from the exponentiated weight. Each part of the formula has a specific role, dynamically linking weight \( W \) to the wingspan \( L \).

When we calculate the formula for a specific weight like 5.2 pounds, culminating in approximately a 4.04 feet wingspan, it shows how changes in \( W \) result in predictable variations in \( L \).

• The power, or exponent, defines sensitivity—how quickly \( L \) changes as \( W \) shifts.
• The constant 2.43 ensures the final wingspan is realistic according to studied data.

Grasping formula interpretation enables one to model predictions intuitively, an integral skill across scientific and engineering sectors.