You Are Working Part-time For An Electronics Company While Attending High School. The Following Table Shows The Hourly Wage, $w(t)$, In Dollars, That You Earn As A Function Of Time, $t$. Time Is Measured In Years Since The

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Introduction

As a high school student working part-time for an electronics company, it's essential to understand how your hourly wage changes over time. The table provided below shows the hourly wage, w(t)w(t), in dollars, as a function of time, tt. Time is measured in years since the start of your employment. In this article, we will delve into the relationship between time and hourly wage, exploring the mathematical concepts that govern this relationship.

The Table: Hourly Wage as a Function of Time

Time (years) Hourly Wage (w(t)w(t))
0 10
1 12
2 15
3 18
4 22
5 25

Analyzing the Data

At first glance, the table suggests that the hourly wage increases as time progresses. However, to gain a deeper understanding of this relationship, we need to examine the data more closely. Let's consider the changes in hourly wage over each year.

  • Year 0-1: The hourly wage increases from $10 to $12, a 20% increase.
  • Year 1-2: The hourly wage increases from $12 to $15, a 25% increase.
  • Year 2-3: The hourly wage increases from $15 to $18, a 20% increase.
  • Year 3-4: The hourly wage increases from $18 to $22, a 22.22% increase.
  • Year 4-5: The hourly wage increases from $22 to $25, a 13.64% increase.

Identifying Patterns and Trends

Upon closer inspection, we notice that the percentage increase in hourly wage is not constant over time. However, there is a discernible pattern: the percentage increase is higher in the early years of employment and decreases as time progresses. This suggests that the hourly wage grows at a faster rate initially and then slows down.

Mathematical Modeling

To better understand this relationship, let's attempt to model the hourly wage as a function of time using mathematical equations. We can start by assuming that the hourly wage grows at a rate proportional to the current wage. This is a common assumption in mathematical modeling, known as the proportional growth model.

Let w(t)w(t) be the hourly wage at time tt. We can write the proportional growth model as:

w(t)=w0⋅(1+r)tw(t) = w_0 \cdot (1 + r)^t

where w0w_0 is the initial hourly wage, rr is the growth rate, and tt is time in years.

Fitting the Model to the Data

To determine the values of w0w_0 and rr, we can use the data from the table. Let's start by finding the initial hourly wage, w0w_0. We know that w(0)=10w(0) = 10, so we can substitute this value into the equation:

10=w0⋅(1+r)010 = w_0 \cdot (1 + r)^0

Simplifying the equation, we get:

w0=10w_0 = 10

Now that we have found w0w_0, we can use the data from the table to find the growth rate, rr. Let's consider the change in hourly wage from year 0 to year 1:

w(1)=12=10⋅(1+r)1w(1) = 12 = 10 \cdot (1 + r)^1

Simplifying the equation, we get:

(1+r)=1.2(1 + r) = 1.2

Solving for rr, we get:

r=0.2r = 0.2

Evaluating the Model

Now that we have found the values of w0w_0 and rr, we can evaluate the model by comparing its predictions with the actual data. Let's start by calculating the hourly wage at each year using the model:

  • Year 0: w(0)=10w(0) = 10
  • Year 1: w(1)=10⋅(1+0.2)1=12w(1) = 10 \cdot (1 + 0.2)^1 = 12
  • Year 2: w(2)=10⋅(1+0.2)2=14.4w(2) = 10 \cdot (1 + 0.2)^2 = 14.4
  • Year 3: w(3)=10⋅(1+0.2)3=17.28w(3) = 10 \cdot (1 + 0.2)^3 = 17.28
  • Year 4: w(4)=10⋅(1+0.2)4=20.736w(4) = 10 \cdot (1 + 0.2)^4 = 20.736
  • Year 5: w(5)=10⋅(1+0.2)5=24.8832w(5) = 10 \cdot (1 + 0.2)^5 = 24.8832

Comparing these predictions with the actual data, we see that the model is a good fit, with some minor discrepancies.

Conclusion

In this article, we explored the relationship between time and hourly wage using mathematical modeling. We identified a pattern in the data, where the percentage increase in hourly wage is higher in the early years of employment and decreases as time progresses. We then used the proportional growth model to fit the data and found that the hourly wage grows at a rate proportional to the current wage. The model provided a good fit to the data, with some minor discrepancies. This analysis demonstrates the importance of mathematical modeling in understanding complex relationships and making informed decisions.

Future Directions

This analysis can be extended in several ways. For example, we can investigate the impact of other factors, such as experience or education level, on the hourly wage. We can also explore the use of more advanced mathematical models, such as the exponential growth model, to better capture the relationship between time and hourly wage. Additionally, we can apply this analysis to other real-world scenarios, such as stock prices or population growth, to gain a deeper understanding of the underlying mathematical relationships.

Introduction

In our previous article, we explored the relationship between time and hourly wage using mathematical modeling. We identified a pattern in the data, where the percentage increase in hourly wage is higher in the early years of employment and decreases as time progresses. We then used the proportional growth model to fit the data and found that the hourly wage grows at a rate proportional to the current wage. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the proportional growth model, and how does it relate to the hourly wage?

A: The proportional growth model is a mathematical equation that describes how a quantity grows over time. In the context of the hourly wage, the model assumes that the wage grows at a rate proportional to the current wage. This means that the wage increases by a fixed percentage each year, rather than by a fixed amount.

Q: How do I calculate the hourly wage using the proportional growth model?

A: To calculate the hourly wage using the proportional growth model, you need to know the initial hourly wage, the growth rate, and the time in years. The formula for the proportional growth model is:

w(t)=w0⋅(1+r)tw(t) = w_0 \cdot (1 + r)^t

where w(t)w(t) is the hourly wage at time tt, w0w_0 is the initial hourly wage, rr is the growth rate, and tt is time in years.

Q: What is the growth rate, and how do I determine it?

A: The growth rate is the rate at which the hourly wage increases each year. To determine the growth rate, you can use the data from the table and calculate the percentage increase in hourly wage over each year. For example, if the hourly wage increases from $10 to $12 over a year, the growth rate is 20%.

Q: Can I use the proportional growth model to predict the hourly wage in the future?

A: Yes, you can use the proportional growth model to predict the hourly wage in the future. However, it's essential to note that the model assumes that the growth rate remains constant over time. If the growth rate changes, the model may not accurately predict the hourly wage.

Q: How does the proportional growth model compare to other mathematical models, such as the exponential growth model?

A: The proportional growth model and the exponential growth model are both used to describe how a quantity grows over time. However, the exponential growth model assumes that the growth rate is constant, but the quantity grows at an accelerating rate. In contrast, the proportional growth model assumes that the growth rate is constant, but the quantity grows at a constant rate.

Q: Can I apply the proportional growth model to other real-world scenarios, such as stock prices or population growth?

A: Yes, you can apply the proportional growth model to other real-world scenarios, such as stock prices or population growth. However, it's essential to note that the model assumes that the growth rate remains constant over time. If the growth rate changes, the model may not accurately predict the outcome.

Q: What are some limitations of the proportional growth model?

A: Some limitations of the proportional growth model include:

  • The model assumes that the growth rate remains constant over time.
  • The model does not account for external factors that may affect the growth rate.
  • The model may not accurately predict the outcome if the growth rate changes.

Conclusion

In this article, we addressed some of the most frequently asked questions related to the relationship between time and hourly wage. We discussed the proportional growth model, how to calculate the hourly wage using the model, and the limitations of the model. We also explored the application of the model to other real-world scenarios and the importance of considering external factors that may affect the growth rate.