Select The Correct Answer.The Variable $p$ Varies Directly As The Square Of $q$. If $ P = 24 P = 24 P = 24 [/tex] When $q = 4$, What Is The Value Of $p$ When $ Q = 10 Q = 10 Q = 10 [/tex]?A. 150 B. 60 C.

Introduction

In mathematics, direct variation is a relationship between two variables where one variable is a constant multiple of the other. This means that as one variable increases or decreases, the other variable also increases or decreases at a constant rate. In this article, we will explore the concept of direct variation and its applications, particularly in the context of the given problem.

Understanding Direct Variation

Direct variation is often represented by the equation:

$$p = kq^2$$

where $p$ is the dependent variable, $q$ is the independent variable, and $k$ is the constant of variation. The constant $k$ is a measure of the rate at which $p$ changes with respect to $q$. In other words, it represents the slope of the line that represents the relationship between $p$ and $q$.

Given Problem

The problem states that the variable $p$ varies directly as the square of $q$. This means that we can represent the relationship between $p$ and $q$ using the equation:

$$p = kq^2$$

We are given that $p = 24$ when $q = 4$. Using this information, we can find the value of the constant $k$.

Finding the Constant of Variation

To find the value of $k$, we can substitute the given values of $p$ and $q$ into the equation:

$$24 = k(4)^2$$

Simplifying the equation, we get:

$$24 = k(16)$$

Dividing both sides by 16, we get:

$$k = \frac{24}{16}$$

Simplifying further, we get:

$$k = \frac{3}{2}$$

Using the Constant of Variation to Find the Value of $p$

Now that we have found the value of $k$, we can use it to find the value of $p$ when $q = 10$. Substituting $k = \frac{3}{2}$ and $q = 10$ into the equation, we get:

$$p = \frac{3}{2}(10)^2$$

Simplifying the equation, we get:

$$p = \frac{3}{2}(100)$$

Multiplying both sides by $\frac{3}{2}$, we get:

$$p = 150$$

Conclusion

In this article, we explored the concept of direct variation and its applications. We used the given problem to find the value of the constant of variation $k$ and then used it to find the value of $p$ when $q = 10$. The final answer is $\boxed{150}$.

Discussion

This problem is a classic example of direct variation. The relationship between $p$ and $q$ is represented by the equation $p = kq^2$, where $k$ is the constant of variation. The value of $k$ is found by substituting the given values of $p$ and $q$ into the equation. Once the value of $k$ is found, it can be used to find the value of $p$ for any given value of $q$.

Real-World Applications

Direct variation has many real-world applications. For example, the relationship between the distance traveled by a car and the time it takes to travel that distance is a direct variation. The relationship between the amount of money earned by an employee and the number of hours worked is also a direct variation.

Practice Problems

1. The variable $y$ varies directly as the square of $x$. If $y = 36$ when $x = 3$, what is the value of $y$ when $x = 6$?
2. The variable $z$ varies directly as the cube of $w$. If $z = 27$ when $w = 3$, what is the value of $z$ when $w = 6$?

Answer Key

1. $y = 108$
2. $z = 729$
   Direct Variation Q&A =====================

Frequently Asked Questions

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is a constant multiple of the other. This means that 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 is often represented by the equation:

$$p = kq^2$$

where $p$ is the dependent variable, $q$ 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 measure of the rate at which $p$ changes with respect to $q$. It represents the slope of the line that represents the relationship between $p$ and $q$.

Q: How do I find the constant of variation?

A: To find the constant of variation, you can substitute the given values of $p$ and $q$ into the equation and solve for $k$.

Q: What is the relationship between the constant of variation and the variables?

A: The constant of variation, $k$, is a constant that multiplies the independent variable, $q$, to produce the dependent variable, $p$.

Q: Can direct variation be represented by other equations?

A: Yes, direct variation can be represented by other equations, such as:

$$p = kq$$

or

$$p = kq^n$$

where $n$ is a constant.

Q: What are some real-world applications of direct variation?

A: Direct variation has many real-world applications, such as:

• The relationship between the distance traveled by a car and the time it takes to travel that distance
• The relationship between the amount of money earned by an employee and the number of hours worked
• The relationship between the cost of a product and the quantity produced

Q: How do I determine if a relationship is a direct variation?

A: To determine if a relationship is a direct variation, you can:

• Graph the relationship and look for a straight line
• Check if the relationship can be represented by an equation of the form $p = kq^2$
• Check if the relationship has a constant rate of change

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:

• Failing to identify the independent and dependent variables
• Failing to find the constant of variation
• Failing to check if the relationship is a direct variation

Q: How do I solve problems involving direct variation?

A: To solve problems involving direct variation, you can:

• Substitute the given values into the equation and solve for the unknown variable
• Use the constant of variation to find the value of the dependent variable
• Check if the relationship is a direct variation and adjust your solution accordingly

Q: What are some tips for mastering direct variation?

A: Some tips for mastering direct variation include:

• Practice, practice, practice!
• Start with simple problems and gradually move on to more complex ones
• Use visual aids, such as graphs and charts, to help you understand the relationship between the variables
• Check your work carefully to avoid mistakes