Problem 34 - Applied Calculus

Question

An Australian \(^{24}\) study found that, if other factors are constant (education, experience, etc.), taller people receive higher wages for the same work. The study reported a "height premium" for men of \(3 \%\) of the hourly wage for a \(10 \mathrm{cm}\) increase in height; for women the height premium reported was \(2 \% .\) We assume that hourly wages are a linear function of height, with slope given by the height premium at the average hourly wage for that gender. (a) The average hourly wage ^{25}\( for a 178 cm Australian man is AU\$29.40. Express the average hourly wage of an Australian man as a function of his height, \)x\( \)\mathrm{cm}\( (b) The average hourly wage for a 164 cm Australian woman is AU 24.78. Express the average hourly wage of an Australian woman as a function of her height, \)y \mathrm{cm}\( (c) What is the difference in average hourly wages between men and women of height \)178 \mathrm{cm} ?$ (d) Is there a height for which men and women are predicted to have the same wage? If so, what is it?

Step-by-Step Solution

Answer

(c) The men's wage at 178 cm is higher. (d) Yes, solve the equation for the height that equates both wages.

Step 1: Define the equation for men's wage

A height increase of 10 cm results in a 3% wage increase for men. The average wage for a 178 cm man is AU$29.40. Let \( W_m(x) \) denote the wage for a man with height \( x \) cm. The slope of the wage function is given by \( \frac{3}{100 \times 10}W_m(178) = 0.003 \times 29.40 \). Calculate the slope and use it to find the wage equation using point-slope form: \( W_m(x) = 29.40 + 0.003 \times 29.40 (x - 178) \).

Step 2: Define the equation for women's wage

A height increase of 10 cm results in a 2% wage increase for women. The average wage for a 164 cm woman is AU$24.78. Let \( W_w(y) \) denote the wage for a woman with height \( y \) cm. The slope of the wage function is given by \( \frac{2}{100 \times 10}W_w(164) = 0.002 \times 24.78 \). Calculate the slope and use it to find the wage equation using point-slope form: \( W_w(y) = 24.78 + 0.002 \times 24.78 (y - 164) \).

Step 3: Compute the average wage difference at 178 cm

Use the functions from Step 1 and Step 2 to calculate the wage difference at 178 cm: \( W_m(178) = 29.40 \) and \( W_w(178) = 24.78 + 0.002 \times 24.78 \times (178 - 164) \). Find the wage for a 178 cm woman, then compute \( W_m(178) - W_w(178) \).

Step 4: Determine if a height exists for equal wages

To find if there's a height where men's and women's wages are equal, set \( W_m(x) = W_w(y) \). This gives the equation: \( 29.40 + 0.0882(x - 178) = 24.78 + 0.04956(y - 164) \). Replace \( y \) with \( x \) to find a common height and solve for \( x \).

Key Concepts

Height Premium

The concept of "Height Premium" implies that taller individuals tend to receive higher wages compared to their shorter counterparts. This is based on the idea that each 10 cm increase in height could result in a specific percentage raise in wages, assuming other factors remain constant. In a study from Australia, it was observed that for every 10 cm increase in height, men's hourly wages increased by 3%, while women's increased by 2%.

This notion is ideal for students exploring how certain traits can impact earning potential. It's important to note that height alone doesn't determine income, but it is seen as one factor among many. Understanding height premium helps in analyzing how different variables contribute to economic outcomes. It provides a real-world application for learning about linear functions and their practical implications in economy.

Linear Equation

A linear equation is a mathematical expression that models a relationship with a constant rate of change. It is usually written in the form of \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.

In the context of the problem, linear equations express the average hourly wage as a function of height. The slope here represents the height premium. For instance, the linear equation for men's wages based on their height, \( W_m(x) = 29.40 + 0.0882(x - 178) \), showcases how each additional centimeter over the base height of 178 cm leads to an incremental change in average wages.

Understanding linear equations is crucial in solving real-life problems where relationships between two variables are consistent. It allows students to predict outcomes and understand how changes in one variable might affect another.

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know a point on the line and the slope. This form is represented by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a known point on the line.

In this scenario, we use point-slope form to establish wage equations for both men and women. For example, for a man with an average hourly wage of AU\(29.40 at a height of 178 cm, the equation is set up with the known point \( (178, 29.40) \) and the slope derived from the height premium. Similarly, for women with an average wage of AU\)24.78 at 164 cm, point-slope form helps define their linear wage equation.

Using point-slope form simplifies understanding linear relationships and setting up equations, which is essential in various fields like economics, physics, and statistics. It gives students a methodical way to translate real-world data into a structured formula.

Average Hourly Wage

The "Average Hourly Wage" refers to the typical earnings of an individual per hour worked. In this exercise, it is influenced by height. For a 178 cm Australian man, the average hourly wage is AU$29.40, whereas for a 164 cm Australian woman, it is AU$24.78.

The exercise demonstrates how average hourly wages can be expressed as a function of an individual's height, using linear functions. This serves to illustrate the application of mathematical concepts in socioeconomic studies, helping to explain variations in earnings among different groups.

By analyzing average hourly wages through mathematical equations, students learn to appreciate the significance of various factors affecting wage disparities. This understanding is beneficial not only in academics but also for critical thinking in addressing real-world socioeconomic challenges.