Problem 46 - 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? $$ U \text { and } P \text { are constant; the relationship of } R \text { and } F \text {. } $$

Step-by-Step Solution

Answer

The relationship between \(R\) and \(F\) is a direct variation.

Step 1: Identify the Variable Relationship

Given the formula: \[ R = \frac{UF}{P} \]where \(U\) and \(P\) are constants, determine the relationship between \(R\) (the glomerular filtration rate) and \(F\).

Step 2: Substitute Constants

Replace \(U\) and \(P\) with constants in the formula:\[ R = \frac{kF}{c} \]where \(k = U\) and \(c = P\). This simplifies to: \[ R = k'F \] with \(k' = \frac{k}{c}\).

Step 3: Analyze the Simplified Formula

Notice that the formula \(R = k'F\) is of the form \[ y = kx \]This is the general form of a direct variation, where \(k'\) is a constant.

Step 4: Determine Variation Type

As the simplified formula \(R = k'F\) represents a direct variation, it shows that \(R\) varies directly with \(F\) when \(U\) and \(P\) are constants.

Key Concepts

glomerular filtration rate

The glomerular filtration rate (GFR) is a crucial measurement in kidney function tests. It indicates how well the kidneys filter blood by removing excess wastes and fluids.

The GFR is typically calculated using the formula \( R = \frac{UF}{P} \) where:
- \( R \) = glomerular filtration rate.
- \( U \) = concentration of a substance in the urine.
- \( F \) = urine flow rate.
- \( P \) = concentration of the same substance in the blood plasma.

In many medical scenarios, calculating the GFR helps in diagnosing and managing kidney diseases. It’s vital to understand how these variables interact to ensure accurate assessment and treatment.

Understanding the relationship between these variables sheds light on the kidney's efficiency. For instance, changes in urine flow rate (\( F \)) can directly influence the filtration rate (\( R \)), assuming \( U \) and \( P \) are constant.

direct variation

Direct variation in algebra indicates that two variables are proportional to each other. When one variable increases or decreases, the other does so in the same ratio. This relationship can be expressed as:
\( y = kx \)
where:
- \( y \) = dependent variable.
- \( x \) = independent variable.
- \( k \) = constant of variation.

In the formula \( R = \frac{UF}{P} \), with \( U \) and \( P \) being constants, the relationship simplifies to:
\( R = k'F \) where \( k' = \frac{U}{P} \). This equation now represents a direct variation between \( R \) and \( F \).

It means that the glomerular filtration rate (\( R \)) varies directly with the urine flow rate (\( F \)). If \( F \) increases, \( R \) increases in proportion, and if \( F \) decreases, \( R \) decreases similarly. This direct proportionality simplifies the understanding of how changes in one variable affect the other.

algebraic relationships

Algebraic relationships describe how variables interact within an equation. Understanding these relationships can help solve real-world problems like the GFR in medicine.

The given formula \( R = \frac{UF}{P} \) is a perfect example. Let's break it down:
1. Identify constants and variables.
2. Simplify the equation by substituting constants.
3. Analyze the simplified equation to determine the type of variation.

In this case, we set \( U = constant \) and \( P = constant \), leading to the simplified equation \( R = k'F \). Here, \( k' = \frac{U}{P} \) is a constant, confirming a direct variation between \( R \) and \( F \).

Understanding this direct variation helps in predicting how changes in urine flow rate (\( F \)) will affect the glomerular filtration rate (\( R \)). Algebraic relationships play a critical role in interpreting and applying formulas in various fields, including medicine.