The Graph Of A Function Passes Through The Point { (50, 3.6)$}$, And { Y$}$ Varies Inversely As { X$}$.Find The Value Of { Y$}$ When { X$}$ Is 75.A. Calculate The Value Of { K = Xy$}$.B. Write
The Graph of a Function and Inverse Variation
Understanding Inverse Variation
Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. This relationship can be represented by the equation y = k/x, where y is the dependent variable, x is the independent variable, and k is a constant. In this article, we will explore how to find the value of y when x is 75, given that the graph of a function passes through the point (50, 3.6) and y varies inversely as x.
The Equation of Inverse Variation
The equation of inverse variation is y = k/x, where k is a constant. To find the value of k, we can use the given point (50, 3.6) and substitute it into the equation. This gives us:
3.6 = k/50
To solve for k, we can multiply both sides of the equation by 50:
k = 3.6(50) k = 180
Calculating the Value of k
Now that we have found the value of k, we can use it to find the value of y when x is 75. We can substitute x = 75 and k = 180 into the equation y = k/x:
y = 180/75 y = 2.4
Writing the Equation of Inverse Variation
Now that we have found the value of k, we can write the equation of inverse variation as:
y = 180/x
The Graph of a Function and Inverse Variation
The graph of a function that varies inversely as x is a hyperbola. A hyperbola is a type of curve that has two branches that open in opposite directions. The graph of the function y = 180/x is a hyperbola that opens to the right.
Finding the Value of y When x is 75
To find the value of y when x is 75, we can substitute x = 75 into the equation y = 180/x:
y = 180/75 y = 2.4
Conclusion
In this article, we have explored how to find the value of y when x is 75, given that the graph of a function passes through the point (50, 3.6) and y varies inversely as x. We have found the value of k to be 180 and used it to write the equation of inverse variation as y = 180/x. We have also found the value of y when x is 75 to be 2.4.
The Importance of Inverse Variation
Inverse variation is an important concept in mathematics and has many real-world applications. It is used to model relationships between variables in fields such as physics, engineering, and economics. Understanding inverse variation is crucial for solving problems in these fields and making predictions about the behavior of complex systems.
Real-World Applications of Inverse Variation
Inverse variation has many real-world applications, including:
- Physics: Inverse variation is used to model the relationship between the force of gravity and the distance between two objects.
- Engineering: Inverse variation is used to model the relationship between the voltage and current in an electrical circuit.
- Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.
Solving Problems with Inverse Variation
To solve problems with inverse variation, you need to follow these steps:
- Write the equation of inverse variation: Write the equation y = k/x, where k is a constant.
- Find the value of k: Use the given point to find the value of k.
- Substitute the value of x: Substitute the value of x into the equation to find the value of y.
- Solve for y: Solve for y to find the value of the dependent variable.
Conclusion
In this article, we have explored how to find the value of y when x is 75, given that the graph of a function passes through the point (50, 3.6) and y varies inversely as x. We have found the value of k to be 180 and used it to write the equation of inverse variation as y = 180/x. We have also found the value of y when x is 75 to be 2.4. Understanding inverse variation is crucial for solving problems in fields such as physics, engineering, and economics.
The Graph of a Function and Inverse Variation
Understanding Inverse Variation
Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. This relationship can be represented by the equation y = k/x, where y is the dependent variable, x is the independent variable, and k is a constant.
The Equation of Inverse Variation
The equation of inverse variation is y = k/x, where k is a constant. To find the value of k, we can use the given point (50, 3.6) and substitute it into the equation.
Calculating the Value of k
Now that we have found the value of k, we can use it to find the value of y when x is 75. We can substitute x = 75 and k = 180 into the equation y = k/x:
y = 180/75 y = 2.4
Writing the Equation of Inverse Variation
Now that we have found the value of k, we can write the equation of inverse variation as:
y = 180/x
The Graph of a Function and Inverse Variation
The graph of a function that varies inversely as x is a hyperbola. A hyperbola is a type of curve that has two branches that open in opposite directions. The graph of the function y = 180/x is a hyperbola that opens to the right.
Finding the Value of y When x is 75
To find the value of y when x is 75, we can substitute x = 75 into the equation y = 180/x:
y = 180/75 y = 2.4
The Importance of Inverse Variation
Inverse variation is an important concept in mathematics and has many real-world applications. It is used to model relationships between variables in fields such as physics, engineering, and economics. Understanding inverse variation is crucial for solving problems in these fields and making predictions about the behavior of complex systems.
Real-World Applications of Inverse Variation
Inverse variation has many real-world applications, including:
- Physics: Inverse variation is used to model the relationship between the force of gravity and the distance between two objects.
- Engineering: Inverse variation is used to model the relationship between the voltage and current in an electrical circuit.
- Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.
Solving Problems with Inverse Variation
To solve problems with inverse variation, you need to follow these steps:
- Write the equation of inverse variation: Write the equation y = k/x, where k is a constant.
- Find the value of k: Use the given point to find the value of k.
- Substitute the value of x: Substitute the value of x into the equation to find the value of y.
- Solve for y: Solve for y to find the value of the dependent variable.
Conclusion
In this article, we have explored how to find the value of y when x is 75, given that the graph of a function passes through the point (50, 3.6) and y varies inversely as x. We have found the value of k to be 180 and used it to write the equation of inverse variation as y = 180/x. We have also found the value of y when x is 75 to be 2.4. Understanding inverse variation is crucial for solving problems in fields such as physics, engineering, and economics.
Q&A: The Graph of a Function and Inverse Variation
Q: What is inverse variation?
A: Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. This relationship can be represented by the equation y = k/x, where y is the dependent variable, x is the independent variable, and k is a constant.
Q: How do I find the value of k in an inverse variation problem?
A: To find the value of k, you can use the given point and substitute it into the equation y = k/x. For example, if the given point is (50, 3.6), you can substitute x = 50 and y = 3.6 into the equation to find the value of k.
Q: What is the equation of inverse variation?
A: The equation of inverse variation is y = k/x, where k is a constant.
Q: How do I write the equation of inverse variation?
A: To write the equation of inverse variation, you need to know the value of k. Once you have the value of k, you can write the equation as y = k/x.
Q: What is the graph of a function that varies inversely as x?
A: The graph of a function that varies inversely as x is a hyperbola. A hyperbola is a type of curve that has two branches that open in opposite directions.
Q: How do I find the value of y when x is 75 in an inverse variation problem?
A: To find the value of y when x is 75, you can substitute x = 75 into the equation y = k/x. For example, if the value of k is 180, you can substitute x = 75 into the equation to find the value of y.
Q: What are some real-world applications of inverse variation?
A: Inverse variation has many real-world applications, including:
- Physics: Inverse variation is used to model the relationship between the force of gravity and the distance between two objects.
- Engineering: Inverse variation is used to model the relationship between the voltage and current in an electrical circuit.
- Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.
Q: How do I solve problems with inverse variation?
A: To solve problems with inverse variation, you need to follow these steps:
- Write the equation of inverse variation: Write the equation y = k/x, where k is a constant.
- Find the value of k: Use the given point to find the value of k.
- Substitute the value of x: Substitute the value of x into the equation to find the value of y.
- Solve for y: Solve for y to find the value of the dependent variable.
Q: What is the importance of inverse variation?
A: Inverse variation is an important concept in mathematics and has many real-world applications. It is used to model relationships between variables in fields such as physics, engineering, and economics. Understanding inverse variation is crucial for solving problems in these fields and making predictions about the behavior of complex systems.
Frequently Asked Questions
Q: What is the difference between inverse variation and direct variation?
A: Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. Direct variation is a relationship between two variables where one variable increases as the other increases.
Q: How do I determine whether a relationship is inverse or direct?
A: To determine whether a relationship is inverse or direct, you can use the following criteria:
- Inverse variation: If one variable increases as the other decreases, the relationship is inverse.
- Direct variation: If one variable increases as the other increases, the relationship is direct.
Q: What is the equation of direct variation?
A: The equation of direct variation is y = kx, where k is a constant.
Conclusion
In this article, we have explored some frequently asked questions about inverse variation. We have discussed how to find the value of k, write the equation of inverse variation, and solve problems with inverse variation. We have also discussed the importance of inverse variation and its real-world applications.