Select The Correct Answer.The Pressure, P P P , With Which Water Passes Through A Pipe Varies Inversely As The Square Of The Pipe's Radius, R 2 R^2 R 2 . If K K K Is The Constant Of Variation, Which Equation Represents This Situation?A.
Introduction
Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. In this scenario, we are dealing with the pressure of water passing through a pipe, which varies inversely as the square of the pipe's radius. This relationship can be represented mathematically using the concept of inverse variation. In this article, we will explore the concept of inverse variation, its application in real-world scenarios, and derive the equation that represents the situation.
What is 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 mathematically using the equation:
y = k/x
where y is the dependent variable, x is the independent variable, and k is the constant of variation.
The Relationship Between Pressure and Pipe Radius
In this scenario, we are dealing with the pressure of water passing through a pipe, which varies inversely as the square of the pipe's radius. This relationship can be represented mathematically using the equation:
p = k/r^2
where p is the pressure of water passing through the pipe, r is the radius of the pipe, and k is the constant of variation.
Deriving the Equation
To derive the equation that represents the situation, we need to understand the concept of inverse variation and its application in real-world scenarios. In this scenario, we are dealing with the pressure of water passing through a pipe, which varies inversely as the square of the pipe's radius. This relationship can be represented mathematically using the equation:
p = k/r^2
where p is the pressure of water passing through the pipe, r is the radius of the pipe, and k is the constant of variation.
The Constant of Variation
The constant of variation, k, is a value that represents the relationship between the two variables. In this scenario, k is a constant that represents the pressure of water passing through the pipe when the radius of the pipe is 1 unit.
Solving for k
To solve for k, we need to rearrange the equation to isolate k. We can do this by multiplying both sides of the equation by r^2:
pr^2 = k
Now, we can solve for k by dividing both sides of the equation by r^2:
k = pr^2
Conclusion
In conclusion, the equation that represents the situation is:
p = k/r^2
where p is the pressure of water passing through the pipe, r is the radius of the pipe, and k is the constant of variation. This equation represents the relationship between the pressure of water passing through a pipe and the square of the pipe's radius.
Real-World Applications
Inverse variation has numerous real-world applications, including:
- Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
- Engineering: Inverse variation is used to describe the relationship between the pressure of a fluid and the area of a pipe.
- Economics: Inverse variation is used to describe the relationship between the price of a good and the quantity demanded.
Example Problems
- A pipe with a radius of 2 cm has a pressure of 1000 Pa. What is the value of k?
- A pipe with a radius of 3 cm has a pressure of 500 Pa. What is the value of k?
Solution
- To solve for k, we need to rearrange the equation to isolate k. We can do this by multiplying both sides of the equation by r^2:
pr^2 = k
Now, we can solve for k by dividing both sides of the equation by r^2:
k = pr^2 k = 1000(2)^2 k = 4000
- To solve for k, we need to rearrange the equation to isolate k. We can do this by multiplying both sides of the equation by r^2:
pr^2 = k
Now, we can solve for k by dividing both sides of the equation by r^2:
k = pr^2 k = 500(3)^2 k = 4500
Conclusion
In conclusion, the equation that represents the situation is:
p = k/r^2
Frequently Asked Questions
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 mathematically using the equation:
y = k/x
where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Q: What is the relationship between pressure and pipe radius?
A: In this scenario, we are dealing with the pressure of water passing through a pipe, which varies inversely as the square of the pipe's radius. This relationship can be represented mathematically using the equation:
p = k/r^2
where p is the pressure of water passing through the pipe, r is the radius of the pipe, and k is the constant of variation.
Q: How do I derive the equation that represents the situation?
A: To derive the equation that represents the situation, you need to understand the concept of inverse variation and its application in real-world scenarios. In this scenario, we are dealing with the pressure of water passing through a pipe, which varies inversely as the square of the pipe's radius. This relationship can be represented mathematically using the equation:
p = k/r^2
where p is the pressure of water passing through the pipe, r is the radius of the pipe, and k is the constant of variation.
Q: What is the constant of variation?
A: The constant of variation, k, is a value that represents the relationship between the two variables. In this scenario, k is a constant that represents the pressure of water passing through the pipe when the radius of the pipe is 1 unit.
Q: How do I solve for k?
A: To solve for k, you need to rearrange the equation to isolate k. You can do this by multiplying both sides of the equation by r^2:
pr^2 = k
Now, you can solve for k by dividing both sides of the equation by r^2:
k = pr^2
Q: What are some real-world applications of inverse variation?
A: Inverse variation has numerous real-world applications, including:
- Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
- Engineering: Inverse variation is used to describe the relationship between the pressure of a fluid and the area of a pipe.
- Economics: Inverse variation is used to describe the relationship between the price of a good and the quantity demanded.
Q: Can you provide some example problems?
A: Here are some example problems:
- A pipe with a radius of 2 cm has a pressure of 1000 Pa. What is the value of k?
- A pipe with a radius of 3 cm has a pressure of 500 Pa. What is the value of k?
Q: How do I solve these example problems?
A: To solve these example problems, you need to rearrange the equation to isolate k. You can do this by multiplying both sides of the equation by r^2:
pr^2 = k
Now, you can solve for k by dividing both sides of the equation by r^2:
k = pr^2
For example problem 1, you can plug in the values to get:
k = 1000(2)^2 k = 4000
For example problem 2, you can plug in the values to get:
k = 500(3)^2 k = 4500
Conclusion
In conclusion, inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. In this article, we have explored the concept of inverse variation, its application in real-world scenarios, and derived the equation that represents the situation. We have also provided some example problems and solutions to help you understand the concept better.