If Y Y Y Varies Directly As X X X And Y Y Y Is 18 When X X X Is 5, Which Expression Can Be Used To Find The Value Of Y Y Y When X X X Is 11?A. Y = 5 18 ( 11 Y = \frac{5}{18}(11 Y = 18 5 ( 11 ]B. Y = 18 5 ( 11 Y = \frac{18}{5}(11 Y = 5 18 ( 11 ]C.
Understanding Direct Variation
Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. In other words, as one variable increases or decreases, the other variable also increases or decreases at a constant rate. This relationship can be represented mathematically as:
y = kx
where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Given Information
We are given that y varies directly as x, and y is 18 when x is 5. This means that we can use the given information to find the value of the constant of variation, k.
Finding the Constant of Variation
To find the value of k, we can substitute the given values of x and y into the equation y = kx.
18 = k(5)
To solve for k, we can divide both sides of the equation by 5.
k = 18/5
k = 3.6
Expression to Find the Value of y
Now that we have found the value of k, we can use it to find the expression that can be used to find the value of y when x is 11.
y = kx
y = (18/5)x
y = (18/5)(11)
Simplifying the Expression
To simplify the expression, we can multiply the numerator and denominator by 11.
y = (18/5)(11)
y = (1811)/(51)
y = 198/5
y = 39.6
Conclusion
In conclusion, the expression that can be used to find the value of y when x is 11 is y = (18/5)(11). This expression represents the direct variation relationship between y and x, where y is 18 when x is 5.
Answer
The correct answer is B. y = (18/5)(11).
Discussion
Direct variation is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. Understanding direct variation and its applications can help us solve problems and make predictions in a wide range of situations.
Examples of Direct Variation
Some examples of direct variation include:
- The distance traveled by a car and the time it takes to travel that distance.
- The amount of money earned by an employee and the number of hours worked.
- The temperature of a substance and the amount of heat energy added to it.
Real-World Applications
Direct variation has numerous real-world applications, including:
- Physics: The motion of objects, the force of gravity, and the energy of particles.
- Engineering: The design of bridges, buildings, and other structures.
- Economics: The relationship between supply and demand, and the price of goods and services.
Conclusion
Frequently Asked Questions
Q: What is direct variation?
A: Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. In other words, as one variable increases or decreases, the other variable also increases or decreases at a constant rate.
Q: How is direct variation represented mathematically?
A: Direct variation can be represented mathematically as:
y = kx
where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Q: What is the constant of variation?
A: The constant of variation, k, is a number that represents the rate at which the dependent variable changes in response to changes in the independent variable.
Q: How do I find the constant of variation?
A: To find the constant of variation, you can use the given information to set up an equation and solve for k. For example, if y is 18 when x is 5, you can set up the equation:
18 = k(5)
To solve for k, you can divide both sides of the equation by 5.
k = 18/5
k = 3.6
Q: What is the equation for direct variation?
A: The equation for direct variation is:
y = kx
where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Q: How do I use the equation for direct variation?
A: To use the equation for direct variation, you can substitute the given values of x and y into the equation and solve for k. Then, you can use the value of k to find the value of y when x is a different value.
Q: What are some examples of direct variation?
A: Some examples of direct variation include:
- The distance traveled by a car and the time it takes to travel that distance.
- The amount of money earned by an employee and the number of hours worked.
- The temperature of a substance and the amount of heat energy added to it.
Q: What are some real-world applications of direct variation?
A: Some real-world applications of direct variation include:
- Physics: The motion of objects, the force of gravity, and the energy of particles.
- Engineering: The design of bridges, buildings, and other structures.
- Economics: The relationship between supply and demand, and the price of goods and services.
Q: How do I determine if a relationship is direct variation?
A: To determine if a relationship is direct variation, you can use the following steps:
- Collect data on the two variables.
- Plot the data on a graph.
- Check if the graph is a straight line.
- If the graph is a straight line, then the relationship is direct variation.
Q: What are some common mistakes to avoid when working with direct variation?
A: Some common mistakes to avoid when working with direct variation include:
- Not checking if the relationship is direct variation before using the equation.
- Not using the correct values of x and y in the equation.
- Not solving for k correctly.
- Not using the correct value of k to find the value of y.
Conclusion
In conclusion, direct variation is a fundamental concept in mathematics that has numerous applications in various fields. Understanding direct variation and its applications can help us solve problems and make predictions in a wide range of situations.