Problem 36 - Algebra A Combined Function

Question

For each statement, find the constant of variation and the variation equation. \(y\) varies inversely as the square of \(x ; y=0.011\) when \(x=10\)

Step-by-Step Solution

Answer

The constant of variation is 1.1, and the variation equation is \(y = \frac{1.1}{x^2}\).

Step 1: Identify the Type of Variation

The problem states that \(y\) varies inversely as the square of \(x\). This means we are dealing with an inverse square variation. The general form for this type of variation is \(y = \frac{k}{x^2}\), where \(k\) is the constant of variation.

Step 2: Use Given Values to Find Constant of Variation

We are given that \(y = 0.011\) when \(x = 10\). Substitute these values into the inverse variation formula: \[0.011 = \frac{k}{10^2}\].

Step 3: Solve for the Constant of Variation \(k\)

Simplify the equation \[0.011 = \frac{k}{100}\]. Multiply both sides by 100 to solve for \(k\): \[k = 0.011 \times 100 = 1.1\].

Step 4: Write the Variation Equation

Now that we have found \(k = 1.1\), substitute \(k\) back into the inverse variation equation to get the specific variation equation: \[y = \frac{1.1}{x^2}\]. This equation describes how \(y\) varies with \(x\).

Key Concepts

Constant of Variation

In inverse square variation problems, the constant of variation is the key component that links the two variables in the variation relationship. When the problem states that one quantity varies inversely with the square of another, it means the product of these two values is a fixed constant. This constant is what we term as the "constant of variation."

To find the constant of variation, we take the specific values given in the problem—here, it's given that when

• The value of y is 0.011
• The value of x is 10

With these values, use the inverse variation formula. Substitute into the equation and solve for the constant, often symbolized as \(k\). Understanding this constant helps you form the foundation of the variation equation that describes the relationship between the given variables.

Variation Equation

The variation equation is a mathematical representation that describes how two quantities change in relation to one another. In the context of inverse square variation, this equation models how one variable decreases as the square of another variable increases.

Once the constant of variation is determined, it becomes part of the variation equation. For inverse square variation, our general equation is \(y = \frac{k}{x^2}\). This formula shows that as the magnitude of \(x\) increases, \(y\) will decrease, provided that \(k\) remains constant.

For example, in the original exercise, once the constant of variation \(k = 1.1\) was found, it gets placed into this formula to create the specific variation equation \(y = \frac{1.1}{x^2}\). This equation can now be used to predict or describe the behavior of \(y\) for any given \(x\) value within the constraints of the problem.

Inverse Variation Formula

The inverse variation formula lies at the heart of problems involving relationships where one quantity decreases as the other increases. Specifically, in inverse square variation, the formula \(y = \frac{k}{x^2}\) illustrates a unique kind of inverse relationship compared to simple inverse variation.

In general, the inverse variation formula serves to directly link the variable \(y\) to the square of \(x\) through the constant \(k\). Understanding and using this formula helps decode how changes in one variable reflect changes in another, albeit in this case, squared and inverted.

• The formula highlights how doubling the value of \(x\) leads to a decrease in \(y\) by a factor of four, signifying its squared inverse property.
• This is what gives it the distinctive power to model many real-world phenomena, such as the intensity of light over distance.

The elegance of the inverse variation formula lies in its simplicity, yet powerful enough to unravel complex relationships in various fields of study.