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Chapter 7: Problem 53

For exercises 53-56, the formula \(F=\frac{100 S_{u} C_{p}}{S_{p} C_{u}}\) describes the fractional excretion of sodium, \(F\). Is the relationship of the given variables a direct variation or an inverse variation? $$ S_{u}, S_{p} \text {, and } C_{u} \text { are constant; the relationship of } F \text { and } C_{p} \text {. } $$

Short Answer

Expert verified
Direct variation

Step by step solution

01

Identify Constants and Variables

Identify the constants and the variables in the given formula. According to the exercise, the constants are: - Constants: - \(S_{u}\) - \(S_{p}\) - \(C_{u}\) - Variables: - \(F\) - \(C_{p}\)
02

Write Down the Given Formula

The formula given is: \[ F = \frac{100 S_{u} C_{p}}{S_{p} C_{u}} \]
03

Simplify the Formula

Since \(S_{u}, S_{p}\), and \(C_{u}\) are constants, simplify the equation to show the relationship between the variables \(F\) and \(C_{p}\): \[ F = \frac{k C_{p}}{1} \] where \(k = \frac{100 S_{u}}{S_{p} C_{u}}\).
04

Determine the Type of Variation

Since \(F\) is proportional to \(C_{p}\) multiplied by a constant \(k\), the relationship can be classified as direct variation. In direct variation, when one variable increases, the other variable also increases proportionally.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant and Variable Identification
In algebra, identifying constants and variables is crucial for understanding and simplifying expressions. Constants are values that do not change. Variables, on the other hand, represent values that can change.

For instance, in the given exercise, the formula is:
\[ F = \frac{100 S_{u} C_{p}}{S_{p} C_{u}} \]
Here, we need to identify which terms are constants and which are variables. According to the exercise:
  • Constants: \( S_{u} \), \( S_{p} \), \( C_{u} \)
  • Variables: \( F \), \( C_{p} \)
Once the constants and variables are identified, it becomes easier to understand their roles in the equation.
Direct Variation
Direct variation occurs when two variables maintain a consistent ratio. That is, as one variable increases, the other variable increases proportionally. Conversely, if one decreases, the other decreases proportionally.

In the formula \[ F = \frac{100 S_{u} C_{p}}{S_{p} C_{u}} \]
since \( S_{u}, S_{p} \), and \( C_{u} \) are constants, we can simplify the formula by combining these constants into one constant \( k \).
This results in:
\[ F = k C_{p} \]
where \[ k = \frac{100 S_{u}}{S_{p} C_{u}} \]
This simplified equation shows that \( F \) and \( C_{p} \) maintain a direct variation. When \( C_{p} \) increases, \( F \) increases proportionally and vice versa. Understanding direct variation helps in predicting the behavior of variables in algebraic expressions.
Algebraic Simplification
Simplifying algebraic expressions is a fundamental process in algebra. It involves reducing an expression to its simplest form to make it easier to work with.
For the given formula, it starts as:
\[ F = \frac{100 S_{u} C_{p}}{S_{p} C_{u}} \]
Given \( S_{u}, S_{p} \), and \( C_{u} \) are constants, we combined them into a single constant \( k \). This transforms the formula to:
\[ F = k C_{p} \]
where \[ k = \frac{100 S_{u}}{S_{p} C_{u}} \]
By simplifying the formula, you can more easily observe the relationship between the variables. Instead of dealing with multiple constants, you're left with a clear, direct variation equation:
  • \( F \) depends directly on \( C_{p} \)
  • Changes in \( C_{p} \) lead to proportional changes in \( F \)
Simplification in algebra helps to clarify complex relationships and make analytical work more straightforward.

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Most popular questions from this chapter

When a student with math anxiety is given a test, feelings of anxiety and panic can make the student feel that he or she cannot do a single problem on the test. What do you think a student should do if this happens?

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

The relationship of the time a tour guide works, \(x\), and the cost to hire the tour guide, \(y\), is a direct variation. When a tour guide works for \(15 \mathrm{hr}\), the cost is \(\$ 1125\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost to hire a tour guide for \(8 \mathrm{hr}\). d. What does \(k\) represent in this equation?

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ A \text { is constant; the relationship of } R \text { and } T \text {. } $$

For exercises 87-90, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(\frac{11}{x}+\frac{13}{12}=1\) Incorrect Answer: Least common denominator is \(12 x\). $$ \begin{aligned} 12 x\left(\frac{11}{x}+\frac{13}{12}\right) &=1 \\ 12 x\left(\frac{11}{x}\right)+12 x\left(\frac{13}{12}\right) &=1 \\ 132+13 x &=1 \\ \frac{-132}{0+13 x} &=-131 \\ \frac{13 x}{13} &=\frac{-131}{13} \\ x &=-\frac{131}{13} \end{aligned} $$
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