Question

write a general formula to describe each variation. \(y\) varies inversely with \(\sqrt{x} ; \quad y=4\) when \(x=9\)

Short answer

The general formula is \( y = \frac{12}{\sqrt{x}} \).

Step-by-step solution

Understand Inverse Variation

When a variable varies inversely with another, it means that as one variable increases, the other decreases proportionally. In mathematical terms, if y varies inversely with \sqrt{x}, then \( y = \frac{k}{\sqrt{x}} \), where \( k \) is a constant.

Find the Constant of Variation

We need to find the value of the constant \( k \). Use the given values: \( y=4 \) when \( x=9 \). Substitute these values into the formula: \( 4 = \frac{k}{\sqrt{9}} \). Simplify this to get \( 4 = \frac{k}{3} \). Multiply both sides by 3 to solve for \( k \): \( k = 12 \).

Write the General Formula

Now that \( k=12 \), substitute \( k \) back into the original inverse variation formula: \( y = \frac{12}{\sqrt{x}} \). This is the general formula describing the variation.

Key concepts

Inverse Variation Formula

Inverse variation is a relationship between two variables where one variable increases while the other decreases. To express this mathematically, we use the inverse variation formula. If we say that variable y varies inversely with the square root of x, we mean that:

\( y = \frac{k}{\sqrt{x}} \) \

Here, k is the constant of variation. It remains the same for all values of x and y associated through this formula. Whenever you see the phrase 'varies inversely,' it is a clue that as one value goes up, the other must go down to keep the product constant.

Verifying an inverse variation:

• If y doubles, then x must decrease in such a way that the product of y and the square root of x remains constant.
• If the square root of x is halved, then y must increase so that their product does not change.

Constant of Variation

The constant of variation, denoted by k, is pivotal in problems involving inverse variation. To figure out the value of k, we use known values of y and x.

Let's revisit the problem: y varies inversely with \(\sqrt{x} \) and we know that y = 4 when x = 9. Plugging these values into the formula, we get:

\( 4 = \frac{k}{\sqrt{9}} \) \

Since \(\sqrt{9} = 3 \), this simplifies to: \( 4 = \frac{k}{3} \)\. Multiply both sides by 3 to solve for k:

\( k = 4 \times 3 \ = 12 \)

This value of k is then used in the general formula. Remember, k always stays constant for the given relationship between y and x. By knowing k, you can figure out y for any value of x and vice versa.

Solving for Constants

Once we have k, we can easily solve for y or x in different scenarios using the inverse variation formula.

For example, with the problem at hand:

Given the constant of variation k = 12 and the inverse variation form \ \( y = \frac{12}{\sqrt{x}} \) \, you can find y for any x by following these steps:

• Choose the value of x you want to find y for.
• Take the square root of the chosen x value.
• Divide the constant k by this square root.

Suppose we want to find y when x = 16:

\(\sqrt{16} = 4 \)
Then \( y = \frac{12}{4} = 3 \).

Following these steps, you can quickly determine y for any x using the formula \( y = \frac{12}{\sqrt{x}} \) \, making it a useful tool for understanding inverse variation in different contexts.