A Quantity M M M Varies Jointly With P P P And R R R . When P = 2 P = 2 P = 2 And R = 4 R = 4 R = 4 , M = 1 M = 1 M = 1 . What Is The Constant Of Variation?A. 1 8 \frac{1}{8} 8 1 B. 1 2 \frac{1}{2} 2 1 C. 2 D. 8

Apr 2, 2025 by ADMIN 217 views

**Understanding Joint Variation**

Joint variation is a type of functional relationship where a quantity varies jointly with two or more other quantities. In other words, if a quantity $m$ varies jointly with $p$ and $r$ , it means that $m$ is directly proportional to both $p$ and $r$ . Mathematically, this can be represented as:

m = k \cdot p \cdot r

where $k$ is the constant of variation.

The Problem

We are given that a quantity $m$ varies jointly with $p$ and $r$ . When $p = 2$ and $r = 4$ , $m = 1$ . We need to find the constant of variation $k$ .

Step 1: Write the Equation of Joint Variation

Since $m$ varies jointly with $p$ and $r$ , we can write the equation of joint variation as:

m = k \cdot p \cdot r

Step 2: Substitute the Given Values

We are given that $p = 2$ , $r = 4$ , and $m = 1$ . We can substitute these values into the equation of joint variation:

1 = k \cdot 2 \cdot 4

Step 3: Solve for the Constant of Variation

To solve for $k$ , we can divide both sides of the equation by $2 \cdot 4$ :

k = \frac{1}{2 \cdot 4}

Step 4: Simplify the Expression

We can simplify the expression for $k$ by dividing the numerator and denominator by their greatest common divisor, which is 1:

k = \frac{1}{8}

Conclusion

Therefore, the constant of variation is $\frac{1}{8}$ .

Why is Joint Variation Important?

Joint variation is an important concept in mathematics because it helps us model real-world relationships between quantities. For example, the cost of a product may vary jointly with the number of units produced and the price per unit. By understanding joint variation, we can make more accurate predictions and decisions in a variety of fields, including business, economics, and engineering.

Real-World Applications of Joint Variation

Joint variation has many real-world applications, including:

Business: The cost of a product may vary jointly with the number of units produced and the price per unit.
Economics: The demand for a product may vary jointly with the price of the product and the income of consumers.
Engineering: The stress on a material may vary jointly with the force applied and the cross-sectional area of the material.

Examples of Joint Variation

Here are some examples of joint variation:

Cost: The cost of a product varies jointly with the number of units produced and the price per unit.
Demand: The demand for a product varies jointly with the price of the product and the income of consumers.
Stress: The stress on a material varies jointly with the force applied and the cross-sectional area of the material.

Solving Joint Variation Problems

To solve joint variation problems, we can follow these steps:

Write the equation of joint variation.
Substitute the given into the equation.
Solve for the constant of variation.
Simplify the expression for the constant of variation.

Common Mistakes to Avoid

When solving joint variation problems, it's easy to make mistakes. Here are some common mistakes to avoid:

Not writing the equation of joint variation: Make sure to write the equation of joint variation before substituting the given values.
Not simplifying the expression: Make sure to simplify the expression for the constant of variation.
Not checking the units: Make sure to check the units of the constant of variation to ensure that they are correct.

Conclusion

Q: What is joint variation?

A: Joint variation is a type of functional relationship where a quantity varies jointly with two or more other quantities. In other words, if a quantity $m$ varies jointly with $p$ and $r$ , it means that $m$ is directly proportional to both $p$ and $r$ .

Q: How do I write the equation of joint variation?

A: To write the equation of joint variation, you need to use the following formula:

m = k \cdot p \cdot r

where $k$ is the constant of variation.

Q: What is the constant of variation?

A: The constant of variation is a number that represents the rate at which the quantity $m$ changes when the quantities $p$ and $r$ change.

Q: How do I find the constant of variation?

A: To find the constant of variation, you need to substitute the given values into the equation of joint variation and solve for $k$ .

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

A: Joint variation has many real-world applications, including:

Business: The cost of a product may vary jointly with the number of units produced and the price per unit.
Economics: The demand for a product may vary jointly with the price of the product and the income of consumers.
Engineering: The stress on a material may vary jointly with the force applied and the cross-sectional area of the material.

Q: How do I solve joint variation problems?

A: To solve joint variation problems, you can follow these steps:

Write the equation of joint variation.
Substitute the given values into the equation.
Solve for the constant of variation.
Simplify the expression for the constant of variation.

Q: What are some common mistakes to avoid when solving joint variation problems?

A: Some common mistakes to avoid when solving joint variation problems include:

Not writing the equation of joint variation: Make sure to write the equation of joint variation before substituting the given values.
Not simplifying the expression: Make sure to simplify the expression for the constant of variation.
Not checking the units: Make sure to check the units of the constant of variation to ensure that they are correct.

Q: Can you give me an example of a joint variation problem?

A: Here's an example of a joint variation problem:

A company produces a product that costs $10 per unit to produce. The cost of the product varies jointly with the number of units produced and the price per unit. If the company produces 100 units and sells them for $20 each, what is the total cost of producing the product?

To solve this problem, you would need to write the equation of joint variation, substitute the given values into the equation, and solve for the constant of variation.

Q: Can you give me another example of a joint variation problem?

A: Here's another example of a joint variation problem:

A material has a stress of 100 pounds per square inch when a force of 10 pounds is applied it. The stress on the material varies jointly with the force applied and the cross-sectional area of the material. If the force applied is increased to 20 pounds, what is the new stress on the material?

To solve this problem, you would need to write the equation of joint variation, substitute the given values into the equation, and solve for the constant of variation.

Conclusion

In conclusion, joint variation is an important concept in mathematics that helps us model real-world relationships between quantities. By understanding joint variation, we can make more accurate predictions and decisions in a variety of fields, including business, economics, and engineering.