Problem 15
Question
For a fixed number of windows, the number of windows washed per hour, \(x\), and the number of hours it takes to wash the windows, \(y\), is an inverse variation. If a person can wash 20 windows per hour, it takes \(9 \mathrm{hr}\) to wash the windows. 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 wash 30 windows per hour, find the time needed to wash the windows.
Step-by-Step Solution
Verified Answer
a. 180 windows. b. \(x \times y = 180\). c. 6 hours.
1Step 1 - Understand the Inverse Variation
Inverse variation means that the product of the two variables is constant. This relationship can be written as: \[ x \times y = k \] where \(x\) is the number of windows washed per hour and \(y\) is the number of hours it takes.
2Step 2 - Identify Given Values
We are given that when \(x = 20\) windows per hour, \(y = 9\) hours.
3Step 3 - Find the Constant of Variation
To find the constant \(k\), use the given values in the inverse variation formula: \[ k = x \times y \] \[ k = 20 \times 9 \] \[ k = 180 \] The unit of \(k\) is windows, because it represents the number of windows to wash.
4Step 4 - Write the Equation
Using the constant of variation \(k = 180\), the equation representing this relationship is: \[ x \times y = 180 \]
5Step 5 - Solve for New Conditions
If a person washes 30 windows per hour, find the time \(y\) needed to wash the windows using the equation: \[ x \times y = 180 \] Substitute \(x = 30\): \[ 30 \times y = 180 \] Solve for \(y\): \[ y = \frac{180}{30} \] \[ y = 6 \] So, it will take 6 hours.
Key Concepts
constant of variationinverse variation equationsolving algebraic equations
constant of variation
The constant of variation, often denoted by the letter 'k', is a crucial component in inverse variation problems. This constant represents a fixed value that the product of two variables will always equal in an inverse relationship. In the context of our window-washing problem, we know that the number of windows washed per hour (\(x\)) and the number of hours (\(y\)) required to wash them are inversely related. This relationship is expressed as:
\[x \times y = k \]
In our problem, we are given that a person can wash 20 windows per hour and it takes them 9 hours to complete the task. Thus, we substitute these values into the equation:
\[k = 20 \times 9 = 180 \]
The constant of variation (\(k\)) here is 180 windows. This means for any given number of windows washed per hour, the product of that number and the hours taken to wash them will always be 180. Understanding the concept of the constant of variation is essential, as it reveals the foundational relationship between the variables involved.
\[x \times y = k \]
In our problem, we are given that a person can wash 20 windows per hour and it takes them 9 hours to complete the task. Thus, we substitute these values into the equation:
\[k = 20 \times 9 = 180 \]
The constant of variation (\(k\)) here is 180 windows. This means for any given number of windows washed per hour, the product of that number and the hours taken to wash them will always be 180. Understanding the concept of the constant of variation is essential, as it reveals the foundational relationship between the variables involved.
inverse variation equation
An inverse variation equation describes a relationship where the product of two variables is constant. This relationship is mathematically represented as:
\[x \times y = k \]
It's essential to understand that as one variable increases, the other must decrease to keep their product constant. In our exercise, the number of windows washed per hour (\(x\)) and the number of hours (\(y\)) it takes to wash them have an inverse relationship.
From our previous section, we found the constant of variation (\(k\)) to be 180. Hence, the inverse variation equation for this problem is:
\[x \times y = 180 \]
This equation allows us to determine the time required to wash the windows at any given rate of washing per hour. For instance, if a person washes 30 windows per hour, we can substitute this value into the equation to find the time (\(y\)):
\[30 \times y = 180 \]
\[\frac{180}{30} = y \]
Solving this, we get \[y = 6 \] hours. Clearly, the inverse variation equation is a practical tool for solving such problems.
\[x \times y = k \]
It's essential to understand that as one variable increases, the other must decrease to keep their product constant. In our exercise, the number of windows washed per hour (\(x\)) and the number of hours (\(y\)) it takes to wash them have an inverse relationship.
From our previous section, we found the constant of variation (\(k\)) to be 180. Hence, the inverse variation equation for this problem is:
\[x \times y = 180 \]
This equation allows us to determine the time required to wash the windows at any given rate of washing per hour. For instance, if a person washes 30 windows per hour, we can substitute this value into the equation to find the time (\(y\)):
\[30 \times y = 180 \]
\[\frac{180}{30} = y \]
Solving this, we get \[y = 6 \] hours. Clearly, the inverse variation equation is a practical tool for solving such problems.
solving algebraic equations
Solving algebraic equations is a fundamental skill in algebra, especially in problems involving inverse variation. Let's break down the steps necessary to isolate the required variable and solve the equation.
Consider the derived inverse variation equation for our window-washing problem:
\[x \times y = 180 \]
Suppose we need to find out the time (\(y\)) it takes for a person washing 30 windows per hour. Here's how we solve it:
1. Substitute the value for the number of windows washed per hour into the equation:
\[30 \times y = 180 \]
2. To isolate \(y\), divide both sides of the equation by 30:
\[ y = \frac{180}{30} \]
3. Simplify the right-hand side to get:
\[ y = 6 \]
Thus, it takes 6 hours.
When solving algebraic equations, follow these general steps:
Consider the derived inverse variation equation for our window-washing problem:
\[x \times y = 180 \]
Suppose we need to find out the time (\(y\)) it takes for a person washing 30 windows per hour. Here's how we solve it:
1. Substitute the value for the number of windows washed per hour into the equation:
\[30 \times y = 180 \]
2. To isolate \(y\), divide both sides of the equation by 30:
\[ y = \frac{180}{30} \]
3. Simplify the right-hand side to get:
\[ y = 6 \]
Thus, it takes 6 hours.
When solving algebraic equations, follow these general steps:
- Understand the problem and identify what you need to find.
- Substitute known values into the equation.
- Isolate the variable by performing inverse operations.
- Simplify to solve for the required variable.
Other exercises in this chapter
Problem 14
For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{5}{7} ; 42 $$
View solution Problem 14
For exercises 7-32, simplify. $$ \left(\frac{r^{2}}{r^{2}+2 r}\right)\left(\frac{r^{2}-4}{r}\right) $$
View solution Problem 15
For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{5 p-5}{4 p+12}}{\frac{10 p+10}{7 p+21}} $$
View solution Problem 15
For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{2}{9 a} ; 27 a^{2} b $$
View solution