Problem 31 - Elementary Algebra

Question

In radiography, a grid reduces the effect of X-ray scattering. The relationship of the interspace distance on the grid, \(x\), and the grid ratio, \(y\), is an inverse variation. When the interspace distance on a grid is 300 micrometers, the grid ratio is 8 . Write an equation that represents this variation. Include the units.

Step-by-Step Solution

Answer

The equation representing the variation is \( y \times x = 2400 \text{ micrometers} \).

Step 1 - Identify the inverse variation formula

In an inverse variation, the relationship between two variables can be expressed as: \[ y \times x = k \]where \(k\) is a constant.

Step 2 - Plug in the given values

Use the given values for \(x\) and \(y\). Here, the interspace distance \(x\) is 300 micrometers and the grid ratio \(y\) is 8. Substitute these values into the inverse variation formula: \[ 8 \times 300 \text{ micrometers} = k \]

Step 3 - Solve for the constant \(k\)

Calculate the value of the constant \(k\): \[ k = 8 \times 300 \text{ micrometers} = 2400 \text{ micrometers} \]

Step 4 - Write the final equation

Now, substitute the value of \(k\) back into the inverse variation formula to write the equation representing the variation: \[ y \times x = 2400 \text{ micrometers} \]

Key Concepts

inverse variation

Inverse variation describes a relationship where one variable increases as the other decreases. This means they move in opposite directions. For example, if the interspace distance on a radiography grid increases, the grid ratio will decrease to maintain balance.

Mathematically, this is shown using the formula: \ \[ y \times x = k \ \]

In this equation, \( y \) and \( x \) are the variables, and \( k \) is a constant. The product of \( y \) and \( x \) always equals \( k \).

Let's consider the radiography grid example. If the interspace distance \( x \) increases, the grid ratio \( y \) must decrease to keep their product equal to \( k \).

algebraic equations

Algebraic equations are mathematical statements that show the equality of two expressions. In our example, we used an inverse variation equation to describe the relationship between grid ratio \( y \) and interspace distance \( x \).

Using the values provided in the problem (300 micrometers for \( x \) and 8 for \( y \)), we can find the constant of variation \( k \).

Substitute these values into the inverse variation formula: \ \[ 8 \times 300 \text{ micrometers} = k \ \]

Solving this equation gives us \( k = 2400 \text{ micrometers} \).

Finally, we can write the complete equation representing the inverse variation as: \ \[ y \times x = 2400 \text{ micrometers} \ \]

This equation shows how any change in \( x \) will inversely affect \( y \) to maintain the constant value \( k \).

constant of variation

The constant of variation \( k \) is an important part of inverse variation equations. It remains the same regardless of the values of \( x \) and \( y \).

In our example, we calculated \( k \) using the given values of grid ratio and interspace distance: \ \[ k = 8 \times 300 \text{ micrometers} = 2400 \text{ micrometers} \ \]

This value of \( k \) helps us understand the relationship between \( x \) and \( y \). No matter what specific values \( x \) and \( y \) take, the product of \( x \) and \( y \) will always equal \( k \).

This constant helps in predicting how changes affect each variable. If the interspace distance \( x \) were to change, we could use our constant \( k \) to find the new grid ratio \( y \) by rearranging the inverse variation equation.