Select The Correct Answer.The Square Of Y Y Y Varies Directly As The Cube Of X X X . When X = 4 X=4 X = 4 , Y = 2 Y=2 Y = 2 . Which Equation Can Be Used To Find Other Combinations Of X X X And Y Y Y ?A. $y^2=\frac{1}{16}

Understanding Direct Variation

Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In this case, we are given that the square of $y$ varies directly as the cube of $x$. This means that $y^2$ is equal to a constant multiple of $x^3$.

The Equation of Direct Variation

The equation of direct variation can be written as:

$$y^2 = kx^3$$

where $k$ is the constant of variation.

Given Information

We are given that when $x=4$, $y=2$. We can use this information to find the value of $k$.

Finding the Constant of Variation

Substituting $x=4$ and $y=2$ into the equation, we get:

$$2^2 = k(4)^3$$

Simplifying, we get:

$$4 = k(64)$$

Dividing both sides by 64, we get:

$$k = \frac{4}{64} = \frac{1}{16}$$

The Equation with the Constant of Variation

Now that we have found the value of $k$, we can substitute it into the equation of direct variation:

$$y^2 = \frac{1}{16}x^3$$

Checking the Answer

The question asks which equation can be used to find other combinations of $x$ and $y$. The equation we derived is:

$$y^2 = \frac{1}{16}x^3$$

This is the same as option A:

$$y^2=\frac{1}{16}x^3$$

Therefore, the correct answer is option A.

Conclusion

In this article, we have seen how to use direct variation to find the equation that relates the square of $y$ to the cube of $x$. We have also seen how to use given information to find the constant of variation and substitute it into the equation. This allows us to find other combinations of $x$ and $y$.

Direct Variation and the Square of y: Key Points

• Direct variation is a relationship between two variables where one variable is a constant multiple of the other.
• The equation of direct variation can be written as $y^2 = kx^3$, where $k$ is the constant of variation.
• We can use given information to find the value of $k$ and substitute it into the equation.
• The equation with the constant of variation can be used to find other combinations of $x$ and $y$.

Direct Variation and the Square of y: Example

Suppose we want to find the value of $y$ when $x=8$. We can use the equation:

$$y^2 = \frac{1}{16}x^3$$

Substituting $x=8$, we get:

$$y^2 = \frac{1}{16}(8)^3$$

Simplifying, we get:

$$y^2 = \frac{1}{16}(512)$$

$$y^2 = 32$$

Taking the square root of both sides, we get:

$$y = \sqrt{32}$$

$$y = 4\sqrt{2}$$

Therefore, when $x=8$, $y=4\sqrt{2}$.

Direct Variation and the Square of y: Real-World Applications

Direct variation has many real-world applications, including:

• Physics: The motion of an object can be described using direct variation.
• Engineering: The stress on a material can be described using direct variation.
• Economics: The demand for a product can be described using direct variation.

In conclusion, direct variation is a powerful tool for describing relationships between variables. By understanding direct variation, we can use it to solve problems and make predictions in a variety of fields.

Direct Variation and the Square of y: Final Thoughts

In this article, we have seen how to use direct variation to find the equation that relates the square of $y$ to the cube of $x$. We have also seen how to use given information to find the constant of variation and substitute it into the equation. This allows us to find other combinations of $x$ and $y$.

Direct variation is a fundamental concept in mathematics, and it has many real-world applications. By understanding direct variation, we can use it to solve problems and make predictions in a variety of fields.

Direct Variation and the Square of y: References

• [1] "Direct Variation." Math Open Reference, mathopenref.com/directvariation.html.
• [2] "Direct Variation." Khan Academy, khanacademy.org/math/algebra/x2f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f6f
  Direct Variation and the Square of y: Q&A =====================================================

Q: What is direct variation?

A: Direct variation is a relationship between two variables where one variable is a constant multiple of the other. In this case, we are given that the square of $y$ varies directly as the cube of $x$.

Q: What is the equation of direct variation?

A: The equation of direct variation can be written as:

$$y^2 = kx^3$$

where $k$ is the constant of variation.

Q: How do we find the constant of variation?

A: We can use given information to find the value of $k$. For example, if we are given that when $x=4$, $y=2$, we can substitute these values into the equation to find $k$.

Q: What is the equation with the constant of variation?

A: Once we have found the value of $k$, we can substitute it into the equation of direct variation to get:

$$y^2 = \frac{1}{16}x^3$$

Q: How do we use the equation to find other combinations of $x$ and $y$?

A: We can use the equation to find other combinations of $x$ and $y$ by substituting different values of $x$ into the equation and solving for $y$.

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

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

• Physics: The motion of an object can be described using direct variation.
• Engineering: The stress on a material can be described using direct variation.
• Economics: The demand for a product can be described using direct variation.

Q: How do we check our answer?

A: We can check our answer by substituting the values of $x$ and $y$ into the equation and making sure that the equation holds true.

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 using the correct equation of direct variation.
• Not finding the correct value of $k$.
• Not checking the answer.

Q: How do we use direct variation to solve problems?

A: We can use direct variation to solve problems by:

• Writing the equation of direct variation.
• Finding the value of $k$.
• Substituting different values of $x$ into the equation and solving for $y$.

Q: What are some tips for working with direct variation?

A: Some tips for working with direct variation include:

• Make sure to use the correct equation of direct variation.
• Find the correct value of $k$.
• Check the answer.

Q: How do we apply direct variation to real-world problems?

A: We can apply direct variation to real-world problems by:

• Identifying the variables involved in the problem.
• Writing the equation of direct variation.
• Finding the value of $k$.
• Substituting different values of $x$ into the equation and solving for $y$.

Q: What are some common applications of direct variation in physics?

A: Some common applications of direct variation in physics include:

• The motion of an object under the influence of gravity.
• The stress on a material.
• The energy of a system.

Q: What are some common applications of direct variation in engineering?

A: Some common applications of direct variation in engineering include:

• The design of bridges and buildings.
• The stress on a material.
• The energy of a system.

Q: What are some common applications of direct variation in economics?

A: Some common applications of direct variation in economics include:

• The demand for a product.
• The supply of a product.
• The price of a product.

Q: How do we use direct variation to model real-world phenomena?

A: We can use direct variation to model real-world phenomena by:

• Identifying the variables involved in the phenomenon.
• Writing the equation of direct variation.
• Finding the value of $k$.
• Substituting different values of $x$ into the equation and solving for $y$.

Q: What are some common challenges when working with direct variation?

A: Some common challenges when working with direct variation include:

• Finding the correct value of $k$.
• Checking the answer.
• Applying direct variation to real-world problems.

Q: How do we overcome these challenges?

A: We can overcome these challenges by:

• Making sure to use the correct equation of direct variation.
• Finding the correct value of $k$.
• Checking the answer.
• Applying direct variation to real-world problems.

Q: What are some common misconceptions about direct variation?

A: Some common misconceptions about direct variation include:

• Thinking that direct variation is only used in physics and engineering.
• Thinking that direct variation is only used to model simple phenomena.
• Thinking that direct variation is only used to find the value of $k$.

Q: How do we correct these misconceptions?

A: We can correct these misconceptions by:

• Providing examples of direct variation in different fields.
• Showing how direct variation can be used to model complex phenomena.
• Emphasizing the importance of finding the correct value of $k$ and checking the answer.