In inverse variation, the constant k can be found by multiplying the two quantities that vary inversely with each other. If y varies inversely as x, then k is simply the product of x and y, i.e., k = xy.
Understanding Inverse Variation and the Constant k
Inverse variation describes a relationship where an increase in one quantity leads to a proportional decrease in another, and vice-versa. This relationship is mathematically expressed as:
$y = \frac{k}{x}$
where:
- y and x are the two quantities that vary inversely.
- k is the constant of proportionality or the constant of variation.
The constant k represents the fixed product of any corresponding pair of x and y values in an inverse variation. Rearranging the formula $y = \frac{k}{x}$ confirms this:
$xy = k$
This means that for any pair of $(x, y)$ values that satisfy an inverse variation, their product will always be equal to k.
Steps to Solve for k
To find the constant k in an inverse variation, follow these simple steps:
- Identify the relationship: Ensure that the problem explicitly states or implies an inverse variation between two quantities.
- Find a corresponding pair of values: Look for a given pair of x and y values (or the equivalent quantities) that are known to vary inversely.
- Multiply the values: Multiply the x-coordinate by the y-coordinate (or the two inversely related quantities). The result will be the constant k.
Example from Reference:
As stated in the provided information, "if y varies inversely as x, and x = 5 when y = 2, then the constant of variation is k = xy = 5(2) = 10." This clearly demonstrates how k is derived from the product of the given values.
Practical Examples and Solutions
Let's look at more examples to solidify the concept.
Example 1: Finding k from Given Points
Suppose the time t it takes to complete a task varies inversely with the number of people p working on it. If it takes 4 hours for 6 people to complete the task, what is the constant of variation k?
Solution:
- Relationship: t varies inversely as p, so $tp = k$.
- Given values: t = 4, p = 6.
- Calculate k: $k = tp = 4 \times 6 = 24$.
So, the constant of variation k is 24. This means that for this specific task, the product of time and the number of people will always be 24.
Example 2: Verifying k Across Multiple Points
Consider a scenario where the volume V of a gas varies inversely with its pressure P at a constant temperature. If $V = 10 \text{ L}$ when $P = 2 \text{ atm}$, we can find k.
- $k = VP = 10 \text{ L} \times 2 \text{ atm} = 20 \text{ L·atm}$
Now, let's see how k remains constant even with different values of V and P:
| Volume (V) in L | Pressure (P) in atm | Product (V * P) = k |
|---|---|---|
| 10 | 2 | 20 |
| 5 | 4 | 20 |
| 2.5 | 8 | 20 |
| 20 | 1 | 20 |
This table illustrates that for any pair of inversely varying quantities, their product consistently yields the constant k.
Importance of the Constant k
The constant k is crucial because it fully defines the specific inverse relationship between the two quantities. Once k is known, you can find any missing value if one of the quantities is given. For instance, if you know $k = 24$ (from Example 1) and you want to find out how many hours it takes 8 people, you can use $t = k/p = 24/8 = 3$ hours.
For more information on inverse variation, you can explore resources like Khan Academy's lessons on direct and inverse variation.
