Problem 32 - Elementary Algebra

Question

When the radiation is constant, the relationship of the current in an X-ray tube, \(x\), and the exposure time, \(y\), is an inverse variation. When the current is 600 milliamp, the exposure time is \(0.2 \mathrm{~s}\). Write an equation that represents this variation. Include the units.

Step-by-Step Solution

Answer

The equation is: \( x \times y = 120 \text{ mA·s} \)

Step 1: Understand the relationship

The problem states that the relationship between the current, x, in an X-ray tube and the exposure time, y, is an inverse variation. This means that as the current increases, the exposure time decreases, and vice versa. Mathematically, we can express this as: \[ x \times y = k \] where k is a constant.

Step 2: Identify known values

We are given that when the current, x, is 600 milliamp, the exposure time, y, is 0.2 seconds. Let's use these values to find the constant k.

Step 3: Calculate the constant k

Substitute the given values into the inverse variation equation: \[ 600 \text{ mA} \times 0.2 \text{ s} = k \] This simplifies to: \[ k = 120 \text{ mA·s} \]

Step 4: Write the equation

Now we know that k = 120 mA·s. The equation that represents the inverse variation is: \[ x \times y = 120 \text{ mA·s} \]

Key Concepts

Inverse Variation

Inverse variation describes a specific type of relationship between two variables. When one variable increases, the other decreases proportionally.
Imagine the seesaw effect: if one side goes up, the other must go down.
In math, this relationship is usually shown with the formula: \( x \times y = k \).
Here, 'x' and 'y' are the variables, and 'k' is a constant value that doesn't change.
For example, in our exercise, the current in an X-ray tube (x) and exposure time (y) show this inverse relationship.
As the current increases, the exposure time decreases. This means they are inversely related.

Constant of Variation

The constant of variation, represented by 'k', is a crucial part of inverse variation.
It is the product of the two variables involved.
In our example, when the current is 600 mA and the exposure time is 0.2 seconds, the product, which is the constant of variation, is 120 mA·s.
This constant helps us set up our inverse variation equation.
Mathematically, we found \( 600 \text{ mA} \times 0.2 \text{ s} = 120 \text{ mA·s} \).
So, our constant 'k' in this scenario is 120 mA·s.
No matter what values you choose for the current (x) and exposure time (y) as long as their product is 120 mA·s, the relationship holds true.

Algebraic Relationships

Algebraic relationships help define how different variables interact with each other.
Inverse variation is one type of these relationships.
Understanding how variables work together can solve many practical problems.
For our X-ray tube problem, knowing the inverse relationship helps set the right exposure times for different currents.
The equation \( x \times y = 120 \) captures the relationship completely.
If you know one value, you can find the other by rearranging the equation.
For example, if you find the current to be 300 mA, the exposure time can be calculated as: \( y = \frac{120 \text{ mA·s}}{300 \text{ mA}} = 0.4 \text{ s} \).
This makes algebraic relationships very useful in practical applications.