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Wages: A worker is reviewing his pay increases over the past several years. The table below shows the hourly wage W, in dollars, that he earned as a function of time t, measured in years since the beginning of 1990. $$ \begin{array}{|c|c|} \hline \text { Time } t & \text { Wage } W \\ \hline 1 & 15.30 \\ \hline 2 & 15.60 \\ \hline 3 & 15.90 \\ \hline 4 & 16.25 \\ \hline \end{array} $$ a. By calculating ratios, show that the data in this table are exponential. (Round the quotients to two decimal places.) b. What is the yearly growth factor for the data? c. The worker can't remember what hourly wage he earned at the beginning of 1990 . Assuming that \(W\) is indeed an exponential function, determine what that hourly wage was. d. Find a formula giving an exponential model for \(W\) as a function of \(t\). e. What percentage raise did the worker receive each year?f. Given that prices increased by 34% over the decade of the 1990s, use your model to determine whether the worker’s wage increases kept pace with inflation.

Step by step solution

01

Determine If Wages Are Exponential

To determine if the wage increases are exponential, calculate the ratio of successive wages:\( \frac{W(2)}{W(1)} = \frac{15.60}{15.30} \approx 1.02 \), \( \frac{W(3)}{W(2)} = \frac{15.90}{15.60} \approx 1.02 \), and \( \frac{W(4)}{W(3)} = \frac{16.25}{15.90} \approx 1.02 \). Since the ratios are consistent, the data suggests the wages are increasing exponentially.

02

Identify Yearly Growth Factor

The common ratio calculated in Step 1 reflects the yearly growth factor. It shows that each year the wages increase by a factor of approximately 1.02.

03

Calculate Initial Hourly Wage

If \(W\) is exponential, use the general formula for exponential growth \(W = W_0 \times (1.02)^t\). At \(t = 1\), \( W = 15.30 = W_0 \times (1.02)^1 \). Solving for \( W_0 \) gives \( W_0 = \frac{15.30}{1.02} \approx 15.00 \). Thus, the initial wage at the start of 1990 was approximately $15.00.

04

Establish Exponential Model Formula

From the initial wage and growth factor, create an equation: \( W = 15.00 \times (1.02)^t \), where \(W\) is the wage and \(t\) is the number of years since 1990.

05

Determine Percentage Raise Per Year

The yearly growth factor is 1.02, meaning a 2% increase per year (since 1.02 means a 2% raise).

06

Compare Wages to Inflation

Calculate the wage at the end of the 1990s using the model with \(t = 10\): \(W = 15.00 \times (1.02)^{10} \approx 18.29\). This is around a 22% increase from $15.00. Inflation increased prices by 34%, which is more than the wage increase.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wage Increase

A wage increase refers to the raise in the amount of money paid to a worker per hour over time. In the exercise, the worker's wages have increased each year since the beginning of 1990. This is evident from the wage table showing increases from $15.30 to $16.25 over four years. To assess whether these wage increments form an exponential sequence, we look at the ratio of successive wages. Calculating these ratios helps us understand that the increases are not just random but follow a specific pattern. Consistency in the ratio between each year's wage points to an exponential model of growth. Ultimately, recognizing a wage increase pattern is crucial in managing personal finance expectations and in comparing it to economic factors like inflation.

Inflation Comparison

Inflation signifies the rate at which prices for goods and services rise, diminishing purchasing power. It's necessary to compare wage increases against inflation to determine if workers' earnings are keeping pace with rising costs. In the problem, wages grew by approximately 2% per year, leading to a total increase of about 22% over ten years. During the same period, prices increased by a total of 34% due to inflation. The analysis shows that while wages did increase, they did not keep up with inflation. This suggests that, despite a nominal wage increase, the worker's real earnings—reflecting purchasing power—actually decreased over the decade. Therefore, comparing wages to inflation is vital for understanding true income growth, as it relates to what one can afford in terms of goods and services.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the context of wages, an exponential function can model predictable growth patterns over time. The formula used here is:\[ W = W_0 \times (1.02)^t \]Where:- \(W\) is the wage at any time \(t\),- \(W_0\) is the initial wage,- and \(1.02\) represents the growth factor.Understanding exponential growth is indispensable for making long-term projections. Unlike linear growth, which adds a fixed amount each period, exponential growth means each period’s growth builds on the prior period’s total, reflecting compound effects.Using such a model helps predict future wages and plan financial strategies accordingly.

Yearly Growth Factor

The yearly growth factor is a numerical representation showing how much a quantity increases each year. In wage analysis, this factor is crucial to understanding how wages build over time. In the problem, the wage growth factor is approximately 1.02. This means each year, the worker's wage increases by about 2% of the previous year's wage. This calculation was made by consistently finding the ratio of wages from consecutive years. This factor is important because it represents the stability of wage growth over time. It also helps in creating exponential models to predict future earnings. Knowing the growth factor allows one to make educated decisions regarding investment in skills or education, aligning wage growth with personal financial plans.