Problem 44
Question
For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } C \text { are constant; the relationship of } R \text { and } T \text {. } $$
Step-by-Step Solution
Verified Answer
The relationship between \(R\) and \(T\) is an inverse variation.
1Step 1: Identify the Formula
The given formula is \( R = \frac{VC}{T} \), which describes the flow rate \(R\) of fluid through an intravenous drip.
2Step 2: Recognize Constants
It is stated that \(V\) and \(C\) are constants.
3Step 3: Express the Formula with Constants
Rewrite the formula by combining \(V\) and \(C\) into a single constant. Let \(K = VC\), then the formula becomes \( R = \frac{K}{T} \).
4Step 4: Analyze the Relationship
The formula \( R = \frac{K}{T} \) indicates that \(R\) and \(T\) have an inverse relationship. As \(T\) increases, \(R\) decreases and vice versa.
Key Concepts
Flow Rate FormulaDirect VariationConstants in EquationsInverse Relationships
Flow Rate Formula
The flow rate formula described in this exercise is crucial in understanding how intravenous drips function. The given formula is: \( R = \frac{VC}{T} \). Here, \(R\) represents the flow rate of the fluid, \(V\) is the volume of the fluid, \(C\) is the concentration of the fluid, and \(T\) is the time.
This formula shows how different factors interact to determine the rate at which fluid enters a patient's bloodstream.
In this context, the volume and concentration are set as constants.
So we can combine them into a single constant, \(K\). This simplifies our formula to: \( R = \frac{K}{T} \).
Breaking down formulas into constants and variables helps us see clear relationships between the factors involved.
This formula shows how different factors interact to determine the rate at which fluid enters a patient's bloodstream.
In this context, the volume and concentration are set as constants.
So we can combine them into a single constant, \(K\). This simplifies our formula to: \( R = \frac{K}{T} \).
Breaking down formulas into constants and variables helps us see clear relationships between the factors involved.
Direct Variation
Direct variation occurs when one variable increases or decreases in direct proportion to another. In simple terms, if one goes up, so does the other, and vice versa.
For a mathematical representation, if \(y\) varies directly as \(x\), the relationship can be shown as: \( y = kx \),
where \(k\) is the constant of variation.
In our exercise, however, the relationship between the flow rate \(R\) and the time \(T\) is not a direct variation. Instead, we have an inverse relationship. Remember, for direct variation, multiplying \(T\) by any factor would mean multiplying \(R\) by the same factor. This is not the case here.
For a mathematical representation, if \(y\) varies directly as \(x\), the relationship can be shown as: \( y = kx \),
where \(k\) is the constant of variation.
In our exercise, however, the relationship between the flow rate \(R\) and the time \(T\) is not a direct variation. Instead, we have an inverse relationship. Remember, for direct variation, multiplying \(T\) by any factor would mean multiplying \(R\) by the same factor. This is not the case here.
Constants in Equations
Constants in equations are fixed values that do not change. They provide a stable reference that helps simplify and solve equations.
In the original flow rate formula, \(V\) (volume) and \(C\) (concentration) are constants.
Combining these constants creates a new constant, \(K = VC\).
This new constant is instrumental in transforming the formula into a simpler form: \( R = \frac{K}{T} \).
Constants help identify the nature of relationships in equations.
By re-writing equations with constants, you can more easily see the connections between variables!
In the original flow rate formula, \(V\) (volume) and \(C\) (concentration) are constants.
Combining these constants creates a new constant, \(K = VC\).
This new constant is instrumental in transforming the formula into a simpler form: \( R = \frac{K}{T} \).
Constants help identify the nature of relationships in equations.
By re-writing equations with constants, you can more easily see the connections between variables!
Inverse Relationships
An inverse relationship means that as one variable increases, the other decreases.
In this exercise, we see this with the formula \( R = \frac{K}{T} \). If time \(T\) increases, flow rate \(R\) decreases, and if \(T\) decreases, \(R\) increases.
Mathematically, inverse variation is shown by the general formula: \( y = \frac{k}{x} \),
where \(y\) is inversely proportional to \(x\) and \(k\) is a constant.
This relationship is crucial in many practical applications, such as how long a resource lasts depending on its consumption rate.
Understanding inverse relationships helps in setting up and interpreting many real-life situations and mathematical problems.
In this exercise, we see this with the formula \( R = \frac{K}{T} \). If time \(T\) increases, flow rate \(R\) decreases, and if \(T\) decreases, \(R\) increases.
Mathematically, inverse variation is shown by the general formula: \( y = \frac{k}{x} \),
where \(y\) is inversely proportional to \(x\) and \(k\) is a constant.
This relationship is crucial in many practical applications, such as how long a resource lasts depending on its consumption rate.
Understanding inverse relationships helps in setting up and interpreting many real-life situations and mathematical problems.
Other exercises in this chapter
Problem 43
For exercises 39-82, simplify. $$ \frac{3 a}{5} \div \frac{9}{10 a^{2}} $$
View solution Problem 43
For exercises 1-66, simplify. $$ \frac{k^{2}-3 k-40}{k^{2}-4 k-45} $$
View solution Problem 44
For exercises 43-58, (a) solve. (b) check. $$ \frac{3}{k}+\frac{7}{18}=\frac{5}{9} $$
View solution Problem 44
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{2}{x-2}+\frac{3}{x+3}}{\frac{2}{x-2}+\frac{1}{x+3}} $$
View solution