Problem 10
Question
Translate each statement of variation into an equation, and use \(k\) as the constant of variation. The volume \((V)\) of a gas varies directly as the absolute temperature \((T)\) and inversely as the pressure \((P)\).
Step-by-Step Solution
Verified Answer
The equation is \( V = \frac{kT}{P} \).
1Step 1: Understand the variation types
In this problem, we have both a direct variation and an inverse variation. Volume \( V \) varies directly with absolute temperature \( T \), and varies inversely with pressure \( P \). This means for direct variation, as \( T \) increases, \( V \) increases, and for inverse variation, as \( P \) increases, \( V \) decreases.
2Step 2: Write the composite variation equation
To express \( V \) in terms of direct and inverse variation, we combine both relationships into one equation. The direct variation with \( T \) means \( V = kT \), where \( k \) is a constant. The inverse variation with \( P \) transforms the equation to \( V = \frac{kT}{P} \).
3Step 3: Combine everything into one equation
The final equation that combines the direct variation of \( V \) with \( T \) and the inverse variation with \( P \) is: \[V = \frac{kT}{P}\] This equation represents how the volume of gas is related to temperature and pressure.
Key Concepts
Direct VariationInverse VariationConstant of Variation
Direct Variation
In mathematics, direct variation describes a relationship where two variables change proportionally. If one variable increases, the other does too, and vice versa. This can be expressed with the equation: \( y = kx \). Here, \( y \) is the variable that directly varies with \( x \), and \( k \) is a non-zero constant known as the constant of variation.
Direct variation implies a linear relationship through the origin of a graph. For every unit increase in \( x \), \( y \) increases by \( k \) units. This relationship can often be seen in real-life situations such as distance versus time at constant speed.
When solving problems involving direct variation, you need to:
Direct variation implies a linear relationship through the origin of a graph. For every unit increase in \( x \), \( y \) increases by \( k \) units. This relationship can often be seen in real-life situations such as distance versus time at constant speed.
When solving problems involving direct variation, you need to:
- Identify the variables that change together.
- Determine the constant of variation \( k \).
- Formulate the direct variation equation \( y = kx \).
Inverse Variation
Inverse variation occurs when one variable increases while another decreases, maintaining a mathematical balance. In other words, if one variable goes up, the other goes down such that their product remains constant. The formula for inverse variation is \( y = \frac{k}{x} \), where \( k \) is a positive constant.
The relationship is non-linear and creates a hyperbolic shape when graphed. It showcases how two variables are inversely linked, affecting each other reciprocally. For instance, the speed of a car and the time it takes to reach a destination represent an inverse variation. The faster the car moves, the less time it takes to reach the destination.
Important points to remember about inverse variation include:
The relationship is non-linear and creates a hyperbolic shape when graphed. It showcases how two variables are inversely linked, affecting each other reciprocally. For instance, the speed of a car and the time it takes to reach a destination represent an inverse variation. The faster the car moves, the less time it takes to reach the destination.
Important points to remember about inverse variation include:
- The product of the variables remains constant (\( xy = k \)).
- Determine the inverse relation by finding \( k \).
- Translate the scenario into an equation with the form \( y = \frac{k}{x} \).
Constant of Variation
The constant of variation is a crucial component in both direct and inverse variation equations. It represents the fixed ratio or product in the relationship between variables.
In direct variation, the constant \( k \) represents how much one variable changes in relation to the change in another. If \( y \) varies directly with \( x \), expressed as \( y = kx \), then \( k \) stays consistent for all levels of \( x \) and \( y \).
In inverse variation, the constant \( k \) signifies the product of the two inversely varying variables. Here, \( y = \frac{k}{x} \) holds true, as long as \( k \) does not change, even though \( x \) and \( y \) do.
Key aspects of the constant of variation include:
In direct variation, the constant \( k \) represents how much one variable changes in relation to the change in another. If \( y \) varies directly with \( x \), expressed as \( y = kx \), then \( k \) stays consistent for all levels of \( x \) and \( y \).
In inverse variation, the constant \( k \) signifies the product of the two inversely varying variables. Here, \( y = \frac{k}{x} \) holds true, as long as \( k \) does not change, even though \( x \) and \( y \) do.
Key aspects of the constant of variation include:
- Identifying what it represents in the equation.
- Ensuring it remains unchanged within the given scenario.
- Using it to solve for unknowns in the variation equation.
Other exercises in this chapter
Problem 9
Graph each of the following linear and quadratic functions. $$f(x)=-x+3$$
View solution Problem 9
Specify the domain and the range for each relation. Also state whether or not the relation is a function. $$\left\\{(x, y) \mid x^{2}=y^{3}\right\\}$$
View solution Problem 10
Determine \((f \circ g)(x)\) and \((g \circ f)(x)\) for each pair of functions. Also specify the domain of \((f \circ g)(x)\) and \((g \circ f)(x)\). (Objective
View solution Problem 10
Graph each of the functions. $$f(x)=\frac{1}{x}-2$$
View solution