What Does $h(40)=1700$ Mean In Terms Of The Problem?$\[ \begin{tabular}{|c|c|} \hline Hours Of Training & Monthly Pay \\ \hline 10 & 1250 \\ \hline 20 & 1400 \\ \hline 30 & 1550 \\ \hline 40 & 1700 \\ \hline 50 & 1850 \\ \hline 60
Introduction
In mathematics, equations are used to represent relationships between variables. When we encounter an equation like $h(40)=1700$, it's essential to understand the context and the variables involved. In this article, we'll delve into the meaning of this equation and explore its significance in a real-world problem.
The Problem Context
The equation $h(40)=1700$ is part of a problem that involves a relationship between hours of training and monthly pay. The problem is likely related to a scenario where an individual's pay is directly proportional to the number of hours they train. The table below provides more information about the relationship between hours of training and monthly pay.
| Hours of Training | Monthly Pay |
|---|---|
| 10 | 1250 |
| 20 | 1400 |
| 30 | 1550 |
| 40 | 1700 |
| 50 | 1850 |
| 60 | - |
Interpreting the Equation
The equation $h(40)=1700$ can be broken down into two parts:
- h: This is a function that represents the relationship between hours of training and monthly pay.
- 40: This is the input value for the function h, representing 40 hours of training.
- 1700: This is the output value of the function h, representing the monthly pay for 40 hours of training.
In other words, the equation states that when an individual trains for 40 hours, their monthly pay is $1700.
Understanding the Function h
The function h is a mathematical representation of the relationship between hours of training and monthly pay. It takes the number of hours trained as input and returns the corresponding monthly pay as output. The function h is likely a linear function, as the relationship between hours of training and monthly pay appears to be directly proportional.
Linear Function
A linear function is a mathematical function that can be represented in the form:
f(x) = mx + b
where:
- m: is the slope of the function, representing the rate of change of the output with respect to the input.
- b: is the y-intercept of the function, representing the output value when the input is zero.
In the case of the function h, the slope (m) represents the rate of change of monthly pay with respect to hours of training. The y-intercept (b) represents the monthly pay when the individual trains for zero hours.
Slope (m)
The slope (m) of the function h can be calculated using the following formula:
m = (y2 - y1) / (x2 - x1)
where:
- y2: is the output value for the input x2.
- y1: is the output value for the input x1.
- x2: is the input value for the output y2.
- x1: is the input value for the output y1.
Using the data from the table, we can calculate the slope (m) as follows:
m = (1700 - 1250) / (40 - 10) m = 450 / 30 m = 15
Y-Intercept (b)
The y-intercept (b) of the function h can be calculated using the following formula:
b = y1 - mx1
where:
- y1: is the output value for the input x1.
- m: is the slope of the function.
- x1: is the input value for the output y1.
Using the data from the table, we can calculate the y-intercept (b) as follows:
b = 1250 - 15(10) b = 1250 - 150 b = 1100
Linear Function Equation
Now that we have calculated the slope (m) and y-intercept (b), we can write the linear function equation as follows:
h(x) = 15x + 1100
This equation represents the relationship between hours of training and monthly pay. When an individual trains for x hours, their monthly pay is given by the equation h(x) = 15x + 1100.
Conclusion
Q: What is the significance of the equation $h(40)=1700$?
A: The equation $h(40)=1700$ represents the relationship between hours of training and monthly pay. It states that when an individual trains for 40 hours, their monthly pay is $1700.
Q: What is the function h?
A: The function h is a mathematical representation of the relationship between hours of training and monthly pay. It takes the number of hours trained as input and returns the corresponding monthly pay as output.
Q: Is the function h a linear function?
A: Yes, the function h is a linear function. It can be represented in the form f(x) = mx + b, where m is the slope of the function and b is the y-intercept.
Q: What is the slope (m) of the function h?
A: The slope (m) of the function h is 15. It represents the rate of change of monthly pay with respect to hours of training.
Q: What is the y-intercept (b) of the function h?
A: The y-intercept (b) of the function h is 1100. It represents the monthly pay when the individual trains for zero hours.
Q: Can you write the linear function equation for the function h?
A: Yes, the linear function equation for the function h is:
h(x) = 15x + 1100
This equation represents the relationship between hours of training and monthly pay.
Q: How can I use the function h to calculate the monthly pay for a given number of hours trained?
A: To calculate the monthly pay for a given number of hours trained, you can plug the value of x into the linear function equation h(x) = 15x + 1100. For example, if you want to calculate the monthly pay for 50 hours trained, you can plug x = 50 into the equation:
h(50) = 15(50) + 1100 h(50) = 750 + 1100 h(50) = 1850
Therefore, the monthly pay for 50 hours trained is $1850.
Q: What are some real-world applications of the function h?
A: The function h has several real-world applications, including:
- Salary calculations: The function h can be used to calculate the salary of an individual based on the number of hours they work.
- Benefits calculations: The function h can be used to calculate the benefits of an individual based on the number of hours they work.
- Budgeting: The function h can be used to create a budget for an individual or a business based on the number of hours worked.
Q: Can I use the function h to calculate the number of hours trained required to earn a certain amount of money?
A: Yes, you can use the function h to calculate the number of hours trained required to earn a certain amount of money. To do this, you can rearrange the linear function equation h(x) = 15x + 1100 to solve for x. For example, if you want to calculate the number of hours trained required to earn $2000, you can set up the equation:
15x + 1100 = 2000
Subtracting 1100 from both sides gives:
15x = 900
Dividing both sides by 15 gives:
x = 60
Therefore, the number of hours trained required to earn $2000 is 60 hours.
Conclusion
In conclusion, the equation $h(40)=1700$ represents the relationship between hours of training and monthly pay. The function h is a linear function that takes the number of hours trained as input and returns the corresponding monthly pay as output. By understanding the function h, we can use it to calculate the monthly pay for a given number of hours trained, as well as the number of hours trained required to earn a certain amount of money.