Problem 45 - 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? $$ F \text { and } P \text { are constant; the relationship of } R \text { and } U \text {. } $$

Step-by-Step Solution

Answer

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

Step 1: Understand the given formula

Start by looking at the given formula: \[ R = \frac{U F}{P} \]Since we are given that both \(F\) and \(P\) are constants, we can simplify the formula to focus on the relationship between \(R\) and \(U\).

Step 2: Simplify the formula

Factor out constants \(F\) and \(P\) from the formula:\[ R = U \cdot \frac{F}{P} \]Let's denote the constant term \(\frac{F}{P}\) as \(k\) (where \(k\) is a constant): \[ R = kU \]

Step 3: Analyze the simplified formula

Now we have:\[ R = kU \]This simplified formula shows that \(R\) varies directly with \(U\) because as \(U\) increases, \(R\) increases proportionally, and vice versa.

Step 4: Determine the type of variation

A direct variation relationship is when one variable is a constant multiple of another. Since \(R\) is a constant multiple of \(U\) (i.e., \(R = kU\)), the relationship between \(R\) and \(U\) is direct.

Key Concepts

glomular filtration rate

The glomular filtration rate (GFR) measures how well your kidneys are filtering blood. It is a crucial indicator of kidney health. The GFR is calculated using the formula \( R = \frac{U F}{P} \), where:

• \( R \) is the glomular filtration rate.
• \( U \) is the concentration of a specific substance in urine.
• \( F \) is the flow rate of urine formation.
• \( P \) is the plasma concentration of that substance.

By understanding these components, you get insights into how well your kidneys are performing their filtering function. This formula allows medical professionals to monitor kidney function and detect issues early on. When F and P are constants, GFR only depends on U, making it easier to interpret kidney performance.

algebraic relationships

Algebraic relationships help us understand the connections between different variables. In the context of GFR, we have the formula \( R = \frac{U F}{P} \). When dealing with algebraic relationships, it's important to know how changes in one variable affect another.
To simplify the expression for GFR, notice that F and P are constants:
\[ R = U \frac{F}{P} = kU \] Here, \( k = \frac{F}{P} \) represents a constant. The simplified form \[ R = kU \] makes it clear that R and U have an algebraic relationship that simplifies understanding how GFR changes with U.

direct proportionality

Direct proportionality occurs when two variables increase or decrease together at a constant rate. In simpler terms, if one variable doubles, the other doubles too. With the GFR formula simplified to \[ R = kU \], it indicates a direct proportionality between R and U.
Direct proportionality is represented as:
\[ y = kx \] For GFR, \[ R = kU \] means that the rate at which kidneys filter blood (R) changes proportionally with the concentration of a substance in urine (U). This relationship helps predict the behavior of one variable given changes in the other.

kidney function

Kidney function is crucial for maintaining overall health. Kidneys filter waste and excess substances from the blood, producing urine. The efficiency of this filtration process is measured by GFR.
Understanding kidney function using GFR involves analyzing how different variables interact. The formula \( R = \frac{U F}{P} \) gives a quantitative measure. Effective kidneys manage to keep R within a healthy range.
Monitoring GFR helps in:

• Identifying kidney diseases early.
• Tracking changes over time.
• Guiding treatment plans.

Ultimately, maintaining good kidney function is vital for preventing complications and managing overall bodily health effectively.