Problem 48 - Elementary Algebra

Question

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { and } P \text { are constant; the relationship of } U \text { and } F \text {. } $$

Step-by-Step Solution

Answer

It's a direct variation between \( U \) and \( F \).

Step 1: Understand the Given Formula

The given formula is \( R = \frac{UF}{P} \), where \(R\), \(U\), \(F\), and \(P\) represent different variables related to the glomular filtration rate by a kidney.

Step 2: Identify Constant Variables

According to the problem, \(R\) and \(P\) are constant. Therefore, we need to examine the relationship between \(U\) and \(F\).

Step 3: Isolate the Relationship of U and F

Rewrite the formula to isolate the relationship between \(U\) and \(F\): \( R = \frac{UF}{P} \). Since \(R\) and \(P\) are constants, let \(C = \frac{P}{R} \), which leads to \( U = CF \).

Step 4: Determine the Type of Variation

Since \(U = CF\), \(U\) is directly proportional to \(F\). This means that if \(F\) increases, \(U\) also increases, and if \(F\) decreases, \(U\) decreases as well.

Key Concepts

glomular filtration rate

The glomular filtration rate (GFR) is a key measure used to evaluate kidney function. It helps determine how well the kidneys are filtering waste from the blood. The formula used in the exercise is: \( R = \frac{UF}{P} \), where:

• \( R \) represents the GFR.
• \( U \) stands for the urine concentration of a substance.
• \( F \) represents the flow rate of urine.
• \( P \) is the plasma concentration of the same substance.

By measuring these variables, doctors can assess whether kidneys are functioning properly. It's crucial for diagnosing and managing kidney diseases.

direct variation

Direct variation describes a relationship where one variable changes directly with another. In our exercise, when we said \( R \) and \( P \) are constants, we looked at the relationship between \( U \) and \( F \). With these constants, the relationship can be simplified as follows: Since \( R \) and \( P \) are constants, let’s rewrite the formula: \( R = \frac{UF}{P} \). To isolate the relationship between \( U \) and \( F \), rewrite it as: \( U = C F \) where \( C = \frac{P}{R} \) is a constant. This equation tells us that \( U \) varies directly with \( F \). What does this mean? It means:

• If \( F \) increases, \( U \) also increases.
• If \( F \) decreases, \( U \) decreases too.

Essentially, \( U \) is directly proportional to \( F \). This can be visually represented on a graph as a straight line passing through the origin.

proportionality

Proportionality is a fundamental concept in mathematics and physics. It describes the relationship between two quantities, where one quantity is a constant multiple of the other. In terms of the GFR formula given, we observe proportionality between \( U \) and \( F \). This relationship can be expressed mathematically as \( U = k F \), where \( k \) is a constant of proportionality. Important points about proportionality:

• The graph of proportional relationships is a straight line through the origin.
• In direct proportionality, as one quantity increases, the other quantity also increases by the same factor.
• The constant of proportionality can give us insight into how much one variable changes in relation to another.

This concept is essential not only in understanding kidney function but also in various scientific analyses and mathematical problem-solving.