Problem 16 - Elementary Algebra

Question

For a fixed number of hotel rooms, the number of rooms cleaned per hour, \(x\), and the number of hours it takes to clean the rooms, \(y\), is an inverse variation. If a person can clean 8 rooms per hour, it takes 15 hr to clean the rooms. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can clean 6 rooms per hour, find the time needed to clean the rooms.

Step-by-Step Solution

Answer

The constant of variation is 120 rooms/hr. The equation is \( x \times y = 120 \). It takes 20 hours to clean at 6 rooms/hr.

Step 1: Understand Inverse Variation

In an inverse variation, the product of the two variables is constant. This means if one variable increases, the other decreases proportionally. The relationship can be represented by the equation: \[ x \times y = k \]where \(k\) is the constant of variation.

Step 1: Find the Constant of Variation

Given that a person can clean 8 rooms per hour, and it takes them 15 hours to clean the rooms, use the equation \[ x \times y = k \]Substitute the given values into the equation: \[ 8 \times 15 = k \]Calculate the product: \[ k = 120 \]Therefore, the constant of variation is 120 rooms per hour \( hr \).

Step 2: Write the Equation

Using the constant of variation \( k = 120 \), substitute into the inverse variation equation: \[ x \times y = 120 \]So, the equation that represents this relationship is: \[ x \times y = 120 \]

Step 3: Find the Time for 6 Rooms per Hour

Now, if a person can clean 6 rooms per hour, we need to find the time required. Use the equation \[ x \times y = 120 \]Substitute \( x = 6 \): \[ 6 \times y = 120 \]Solve for \( y \): \[ y = 120 / 6 \]\[ y = 20 \]So, it takes 20 hours to clean the rooms when cleaning at a rate of 6 rooms per hour.

Key Concepts

constant of variation

In the world of inverse variation, understanding the 'constant of variation,' denoted by the symbol \(k\), is crucial. This constant shows the relationship between two variables where one increases as the other decreases. Think of it like this: the product of these two variables always equals a constant value.

For example, if you have a scenario where the number of rooms cleaned per hour (\(x\)) and the number of hours it takes to clean those rooms (\(y\)) are inversely related, their product remains constant. The formula for inverse variation is:

\[ x \times y = k \]
Consider an instance where a person cleans 8 rooms per hour, and it takes them 15 hours to finish cleaning. By substituting these values into the formula, you get:
\[ 8 \times 15 = k \]
Calculating this, you'll find
\[ k = 120 \]
This tells us that the constant of variation in this example is 120 rooms per hour \(hr\).

inverse variation equation

Once you've identified the constant of variation, you can create the inverse variation equation easily. An inverse variation equation helps in describing the consistent relationship between two variables as one changes. It's represented using the constant of variation \(k\) and can be written as:

\[ x \times y = k \]
In our example, after finding \(k = 120\), the equation becomes:

\[ x \times y = 120 \]
This equation tells us that for any given number of rooms cleaned per hour \(x\), the product of \(x\) and the hours spent cleaning \(y\) will always equal 120. This is a powerful tool because it allows you to find unknown values easily if you have one of the variables. Such equations are frequently encountered in situations involving time, speed, work rates, and more.

time calculation

Now, let's tackle time calculation using the inverse variation equation. Let's say you need to determine the time required to clean the rooms if a person cleans at a different rate. Using our previously established equation \( x \times y = 120 \), we can find the required time for any given cleaning rate.

For instance, if the cleaning rate is 6 rooms per hour, you can set \( x = 6 \) and need to solve for \( y \) in the equation:

\[ 6 \times y = 120 \]
To isolate \(y\), divide both sides by 6:
\[ y = 120 / 6 \]
This gives us:
\[ y = 20 \]
So, it would take 20 hours to clean all the rooms at a rate of 6 rooms per hour. This straightforward calculation is essential, especially in practical applications like scheduling work or managing time effectively. By knowing how to adjust variables, you can plan more efficiently.