$y$ Varies Inversely With $x$, And $y=20$ When $x=15$. What Is The Value Of $y$ When $x=50$?A. $y=19$B. $y=6$C. $y=37$

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Understanding Inverse Variation

Inverse variation is a relationship between two variables, x and y, where the product of the two variables is constant. This relationship can be expressed as:

y=kxy = \frac{k}{x}

where k is a constant.

Given Values

We are given that y varies inversely with x, and y = 20 when x = 15. We can use this information to find the value of the constant k.

Finding the Constant k

To find the value of k, we can substitute the given values into the equation:

20=k1520 = \frac{k}{15}

We can then solve for k by multiplying both sides of the equation by 15:

k=20×15k = 20 \times 15

k=300k = 300

The Equation with k

Now that we have found the value of k, we can write the equation of inverse variation:

y=300xy = \frac{300}{x}

Solving for y when x = 50

We are asked to find the value of y when x = 50. We can substitute x = 50 into the equation:

y=30050y = \frac{300}{50}

We can then simplify the fraction by dividing 300 by 50:

y=6y = 6

Conclusion

Therefore, the value of y when x = 50 is 6.

Answer

The correct answer is B. y = 6.

Discussion

Inverse variation is a fundamental concept in mathematics, and it has many real-world applications. In this problem, we used the concept of inverse variation to solve for y when x is given. We found the value of the constant k using the given values, and then we used the equation of inverse variation to solve for y when x = 50.

Real-World Applications

Inverse variation has many real-world applications, including:

  • Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
  • Engineering: Inverse variation is used to describe the relationship between the voltage and current in an electrical circuit.
  • Economics: Inverse variation is used to describe the relationship between the price of a good and the quantity demanded.

Conclusion

Q: What is inverse variation?

A: Inverse variation is a relationship between two variables, x and y, where the product of the two variables is constant. This relationship can be expressed as:

y=kxy = \frac{k}{x}

where k is a constant.

Q: How do I know if a relationship is an inverse variation?

A: To determine if a relationship is an inverse variation, you can use the following steps:

  1. Graph the relationship: If the graph of the relationship is a hyperbola, then it is an inverse variation.
  2. Check the equation: If the equation of the relationship is of the form y = k/x, then it is an inverse variation.
  3. Check the data: If the data points on the graph are not linear, but rather follow a curved path, then it is an inverse variation.

Q: How do I find the constant k in an inverse variation?

A: To find the constant k in an inverse variation, you can use the following steps:

  1. Use the given values: If you are given two values of x and y, you can substitute them into the equation to find k.
  2. Solve for k: Once you have substituted the values into the equation, you can solve for k by multiplying both sides of the equation by x.
  3. Check your answer: Once you have found k, you can check your answer by substituting it back into the equation and making sure that it is true.

Q: How do I solve for y in an inverse variation?

A: To solve for y in an inverse variation, you can use the following steps:

  1. Substitute the value of x: Once you have found the value of x, you can substitute it into the equation to find y.
  2. Solve for y: Once you have substituted the value of x into the equation, you can solve for y by multiplying both sides of the equation by x.
  3. Check your answer: Once you have found y, you can check your answer by substituting it back into the equation and making sure that it is true.

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

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

  • Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
  • Engineering: Inverse variation is used to describe the relationship between the voltage and current in an electrical circuit.
  • Economics: Inverse variation is used to describe the relationship between the price of a good and the quantity demanded.

Q: How do I graph an inverse variation?

A: To graph an inverse variation, you can use the following steps:

  1. Plot the points: Once you have found the values of x and y, you can plot the points on a graph.
  2. Draw the hyperbola: Once you have plotted the points, you can draw the hyperbola that represents the inverse variation.
  3. Label the axes: Once you have drawn the hyperbola, you can label the axes to show the relationship between x and y.

Q: What are some common mistakes to avoid when working with inverse variation?

A: Some common mistakes to avoid when working with inverse variation include:

  • Not checking the equation: Make sure to check the equation to make sure that it is in the form y = k/x.
  • Not checking the data: Make sure to check the data to make sure that it is not linear, but rather follows a curved path.
  • Not solving for k: Make sure to solve for k by multiplying both sides of the equation by x.

Conclusion

In conclusion, inverse variation is a fundamental concept in mathematics that has many real-world applications. We have discussed the concept of inverse variation, how to find the constant k, how to solve for y, and how to graph an inverse variation. We have also discussed some common mistakes to avoid when working with inverse variation.