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
Understanding the Problem
As a high school student working part-time for an electronics company, you are paid an hourly wage that varies with time. The table below shows the hourly wage, { w(t) $}$, in dollars, that you earn as a function of time, { t $}$. Time is measured in years since the start of your job.
| Time (t) | Hourly Wage (w(t)) |
|---|---|
| 0 | 10 |
| 1 | 12 |
| 2 | 15 |
| 3 | 18 |
| 4 | 20 |
| 5 | 22 |
| 6 | 25 |
| 7 | 28 |
| 8 | 30 |
| 9 | 32 |
| 10 | 35 |
Analyzing the Data
The table shows that your hourly wage increases as time passes. This means that your earnings will increase over time. However, the rate at which your earnings increase is not constant. To understand this better, we need to analyze the data.
Calculating the Rate of Change
The rate of change of your hourly wage with respect to time is given by the derivative of the function w(t). We can calculate this using the following formula:
{ \frac{dw}{dt} = \lim_{h \to 0} \frac{w(t+h) - w(t)}{h} $}$
Using this formula, we can calculate the rate of change of your hourly wage at different times.
| Time (t) | Hourly Wage (w(t)) | Rate of Change (dw/dt) |
|---|---|---|
| 0 | 10 | 2 |
| 1 | 12 | 3 |
| 2 | 15 | 5 |
| 3 | 18 | 7 |
| 4 | 20 | 9 |
| 5 | 22 | 11 |
| 6 | 25 | 13 |
| 7 | 28 | 15 |
| 8 | 30 | 17 |
| 9 | 32 | 19 |
| 10 | 35 | 21 |
Interpreting the Results
The results show that the rate of change of your hourly wage is not constant. It increases over time, which means that your earnings will increase at a faster rate as time passes. This is because the hourly wage is increasing at a faster rate as time passes.
Conclusion
In conclusion, the table shows that your hourly wage increases as time passes. The rate of change of your hourly wage is not constant and increases over time. This means that your earnings will increase at a faster rate as time passes.
Mathematical Analysis
To analyze the data mathematically, we can use the concept of a function. A function is a relation between a set of inputs and a set of possible outputs. In this case, the function w(t) represents the hourly wage as a function of time.
We can represent the function w(t) using a graph. The graph will show the hourly wage as a function of time.
Graphical Representation
The graph of the function w(t) is a straight line with a positive slope. This means that the hourly wage increases as time passes.
Mathematical Representation
We can represent the function w(t) mathematically using the following equation:
{ w(t) = 2t + 10 $}$
This equation represents the hourly wage as a function of time.
Derivative of the Function
The derivative of the function w(t) represents the rate of change of the hourly wage with respect to time. We can calculate the derivative using the following formula:
{ \frac{dw}{dt} = \lim_{h \to 0} \frac{w(t+h) - w(t)}{h} $}$
Using this formula, we can calculate the derivative of the function w(t).
Second Derivative of the Function
The second derivative of the function w(t) represents the rate of change of the rate of change of the hourly wage with respect to time. We can calculate the second derivative using the following formula:
{ \frac{d2w}{dt2} = \lim_{h \to 0} \frac{\frac{dw}{dt}(t+h) - \frac{dw}{dt}(t)}{h} $}$
Using this formula, we can calculate the second derivative of the function w(t).
Conclusion
In conclusion, the mathematical analysis of the data shows that the hourly wage increases as time passes. The rate of change of the hourly wage is not constant and increases over time. This means that your earnings will increase at a faster rate as time passes.
Real-World Applications
The analysis of the data has real-world applications. For example, it can be used to determine the hourly wage of an employee based on their time of service. It can also be used to determine the rate of change of the hourly wage over time.
Conclusion
In conclusion, the analysis of the data shows that the hourly wage increases as time passes. The rate of change of the hourly wage is not constant and increases over time. This means that your earnings will increase at a faster rate as time passes.
Recommendations
Based on the analysis of the data, the following recommendations can be made:
- The hourly wage should be increased at a faster rate as time passes.
- The rate of change of the hourly wage should be monitored regularly to ensure that it is increasing at a rate that is consistent with the company's goals.
- The data should be analyzed regularly to determine the rate of change of the hourly wage and to make recommendations for future increases.
Conclusion
Q&A: Understanding the Problem
Q: What is the problem?
A: The problem is that you are working part-time for an electronics company while attending high school, and your hourly wage varies with time.
Q: What is the hourly wage?
A: The hourly wage is given by the function w(t), where t is the time in years since the start of your job.
Q: What is the rate of change of the hourly wage?
A: The rate of change of the hourly wage is given by the derivative of the function w(t), which is denoted by dw/dt.
Q: How do I calculate the rate of change of the hourly wage?
A: To calculate the rate of change of the hourly wage, you can use the following formula:
{ \frac{dw}{dt} = \lim_{h \to 0} \frac{w(t+h) - w(t)}{h} $}$
Q: What is the significance of the rate of change of the hourly wage?
A: The rate of change of the hourly wage is significant because it determines how fast your earnings will increase over time.
Q: How do I interpret the results of the rate of change of the hourly wage?
A: To interpret the results of the rate of change of the hourly wage, you need to understand that it represents the rate at which your earnings will increase over time.
Q: What are the real-world applications of the analysis of the hourly wage?
A: The analysis of the hourly wage has real-world applications, such as determining the hourly wage of an employee based on their time of service and determining the rate of change of the hourly wage over time.
Q: What are the recommendations for the analysis of the hourly wage?
A: The recommendations for the analysis of the hourly wage are to increase the hourly wage at a faster rate as time passes, to monitor the rate of change of the hourly wage regularly, and to analyze the data regularly to determine the rate of change of the hourly wage and to make recommendations for future increases.
Q: What is the conclusion of the analysis of the hourly wage?
A: The conclusion of the analysis of the hourly wage is that the hourly wage increases as time passes, and the rate of change of the hourly wage is not constant and increases over time.
Q: What are the implications of the analysis of the hourly wage?
A: The implications of the analysis of the hourly wage are that your earnings will increase at a faster rate as time passes, and you should take advantage of this by increasing your work hours or seeking out higher-paying jobs.
Q: What are the limitations of the analysis of the hourly wage?
A: The limitations of the analysis of the hourly wage are that it assumes a linear relationship between the hourly wage and time, and it does not take into account other factors that may affect the hourly wage, such as inflation or changes in the economy.
Q: What are the future directions of the analysis of the hourly wage?
A: The future directions of the analysis of the hourly wage are to develop more sophisticated models that take into account other factors that may affect the hourly wage, and to use data from other sources to validate the results of the analysis.
Conclusion
In conclusion, the analysis of the hourly wage shows that the hourly wage increases as time passes, and the rate of change of the hourly wage is not constant and increases over time. This means that your earnings will increase at a faster rate as time passes.