Understanding Joint Variation

Oct 13, 2025 by miodeya.com 2 views

If ${$ y $}$ varies jointly as ${$ x $}$ and the cube of ${$ z $}$, and ${$ y = 16 $}$ when ${$ x = 4 $}$ and ${$ z = 2 $}$, find ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$

Understanding Joint Variation

Joint variation is a type of variation where a quantity varies directly with the product of two or more other quantities. In this case, we are given that ${$ y $}$ varies jointly as ${$ x $}$ and the cube of ${$ z $}$. This can be expressed mathematically as:

${$ y = kxz^3 $}$

where ${$ k $}$ is the constant of variation.

Given Values

We are given that ${$ y = 16 $}$ when ${$ x = 4 $}$ and ${$ z = 2 $}$. We can use these values to find the constant of variation ${$ k $}$.

Finding the Constant of Variation

Substituting the given values into the equation, we get:

${$ 16 = k(4)(2)^3 $}$

Simplifying the equation, we get:

${$ 16 = k(4)(8) $}$

${$ 16 = 32k $}$

Dividing both sides by 32, we get:

${$ k = \frac{16}{32} $}$

${$ k = \frac{1}{2} $}$

The Equation with the Constant of Variation

Now that we have found the constant of variation, we can write the equation as:

${$ y = \frac{1}{2}xz^3 $}$

Finding ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$

We can substitute the values of ${$ x $}$ and ${$ z $}$ into the equation to find ${$ y $}$:

${$ y = \frac{1}{2}(-8)(-3)^3 $}$

Simplifying the equation, we get:

${$ y = \frac{1}{2}(-8)(-27) $}$

${$ y = \frac{1}{2}(216) $}$

${$ y = 108 $}$

Conclusion

In this article, we have discussed joint variation and how to find the constant of variation. We have also used the given values to find the constant of variation and then used it to find ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$. The final answer is ${$ y = 108 $}$.

Understanding Joint Variation

${$ y = kxz^3 $}$

where ${$ k $}$ is the constant of variation.

Types of Variation

There are several types of variation, including:

Direct Variation: A quantity varies directly with another quantity.
Inverse Variation: A quantity varies inversely with another quantity.
Joint Variation: A quantity varies jointly with two or more other quantities.

Real-World Applications

Joint variation has many real-world applications, including:

Physics: The force of gravity varies jointly with the mass of two objects and the distance between them.
Engineering: The stress on a material varies jointly with the force applied and the cross-sectional area of the material.
Economics: The demand for a product varies jointly with the price of the product and the income of the consumer.

Conclusion

In conclusion, joint variation is a type of variation where a quantity varies directly with the product of two or more other quantities. We have discussed how to find the constant of variation and used it to find ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$. The final answer is ${$ y = 108 $}$.
Q&A: Joint Variation

Understanding Joint Variation

${$ y = kxz^3 $}$

where ${$ k $}$ is the constant of variation.

Frequently Asked Questions

Q: What is joint variation?

A: Joint variation is a type of variation where a quantity varies directly with the product of two or more other quantities.

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 and solve for the constant.

Q: What is the equation for joint variation?

A: The equation for joint variation is:

${$ y = kxz^3 $}$

Q: How do I use the equation to find ${$ y $}$?

A: To use the equation to find ${$ y $}$, you need to substitute the values of ${$ x $}$ and ${$ z $}$ into the equation and solve for ${$ y $}$.

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

A: Joint variation has many real-world applications, including physics, engineering, and economics.

Q: Can you give an example of joint variation in physics?

A: Yes, the force of gravity varies jointly with the mass of two objects and the distance between them.

Q: Can you give an example of joint variation in engineering?

A: Yes, the stress on a material varies jointly with the force applied and the cross-sectional area of the material.

Q: Can you give an example of joint variation in economics?

A: Yes, the demand for a product varies jointly with the price of the product and the income of the consumer.

Q: How do I know if a problem is a joint variation problem?

A: If a problem involves a quantity that varies directly with the product of two or more other quantities, it is likely a joint variation problem.

Q: Can you give a problem that involves joint variation?

A: Yes, if ${$ y $}$ varies jointly as ${$ x $}$ and the cube of ${$ z $}$, and ${$ y = 16 $}$ when ${$ x = 4 $}$ and ${$ z = 2 $}$, find ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$.

Answer

${$ y = 108 $}$

Conclusion

In conclusion, joint variation is a type of variation where a quantity varies directly with the product of two or more other quantities. We have discussed how to find the constant of variation and used it to find ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$. We have also answered some frequently asked questions about joint variation.

Additional Resources

Joint Variation Formula: ${$ y = kxz^3 $}$
Constant of Variation: ${$ k $}$
Joint Variation Problems: Examples of joint variation problems in physics, engineering, and economics.

Conclusion

In conclusion, joint variation is a type of variation where a quantity varies directly with the product of two or more other quantities. We have discussed how to find the constant of variation and used it to find ${$ y $}$ when ${$ x = -8 $}$ and ${$ z = -3 $}$. We have also answered some frequently asked questions about joint variation.