Problem 42 - Algebra A Combined Function

Question

The formula \(w=0.00185 h^{2.67}\) is used to estimate the normal weight w in pounds of a boy h inches tall. Use this formula to solve. Round to the nearest tenth. Find the expected height of a boy who weighs 140 pounds.

Step-by-Step Solution

Answer

The expected height is approximately 67.5 inches.

Step 1: Understanding the Formula

The given formula is \( w = 0.00185 h^{2.67} \), where \( w \) represents the weight in pounds and \( h \) represents the height in inches. We are tasked with finding the height \( h \) when the weight \( w \) is 140 pounds.

Step 2: Set the Equation

Plug the given weight into the formula to form the equation. We have \( 140 = 0.00185 h^{2.67} \). Our goal is to solve for \( h \).

Step 3: Isolate the Exponential Term

Divide both sides of the equation by 0.00185 to isolate the \( h^{2.67} \) term:\[\frac{140}{0.00185} = h^{2.67}\]

Step 4: Calculate the Right-Hand Side

Calculate \( \frac{140}{0.00185} \) to get it to the power format. This gives us approximately 75675.68.\[ h^{2.67} = 75675.68 \]

Step 5: Solve for Height h

To solve for \( h \), take the \( (1/2.67) \) power of both sides:\[ h = 75675.68^{\frac{1}{2.67}} \]

Step 6: Calculate the Height

Compute \( 75675.68^{\frac{1}{2.67}} \). This equals approximately 67.521 inches. Round to the nearest tenth to get \( h \approx 67.5 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of the given problem, the formula involves an exponential function where the height variable, \( h \), is raised to a power of 2.67. Exponential functions appear frequently in mathematical modeling, as they can represent rapid growth or decay patterns.
This type of function is expressed as \( y = a \, b^x \), where:

• \( y \) is the output.
• \( a \) is a constant coefficient.
• \( b \) is the base.
• \( x \) is the exponent.

In our formula \( w = 0.00185 h^{2.67} \), \( w \) acts as the output (the boy's weight), \( 0.00185 \) is a constant multiplier, and \( h \) is the height variable being exponentiated.
Exponential functions are key because they show us how small changes to the exponent variable—in this case, height—can result in significant changes to the outcome (weight), showcasing the sensitivity of exponential growth.

Problem Solving

Problem solving in mathematics often involves breaking down a complex problem into manageable steps. By following a logical progression, one can derive solutions systematically. With the given formula, our aim is to find the height corresponding to a given weight, which requires algebraic manipulation.
Start by clearly understanding and stating the problem: We need to find \( h \) such that \( w = 140 \) using the formula for weight. Then, identify how the given formula relates the known variables to the unknown variable \( h \).
These are the structured steps that were used:

• Set up the equation with known values: \( 140 = 0.00185 h^{2.67} \).
• Isolate the variable by dividing both sides by the constant \( 0.00185 \).
• Compute the result and simplify the equation, making it \( h^{2.67} = 75675.68 \).
• Solve for \( h \) by applying the inverse operation (raising to the power of \( 1/2.67 \)).

This problem-solving approach highlights the importance of balancing and simplification in solving equations, which are fundamental techniques in algebra.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena. In this exercise, the formula \( w = 0.00185 h^{2.67} \) is a model used to estimate the weight of boys based on their height. It translates a complex relationship into a simplified mathematical form.
By using such models, predictions and estimations become possible without extensive data collection. Here's why this methodology works well:

• It captures the non-linear nature of growth using an exponential component.
• Provides a generalized relationship applicable to a wide range of heights.
• Simplifies the estimation process for practical applications.

Mathematical modeling requires assumptions and approximations—for instance, the equation assumes a consistent growth pattern and applies a single power throughout. Nonetheless, it's a powerful tool for interpreting and predicting outcomes effectively in real-world scenarios.