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Chapter 2: Problem 43

Explain what is meant by combined variation. Give an example with your explanation.

Short Answer

Expert verified
Combined variation occurs when a quantity varies directly as the product or quotient of two or more other quantities. An example is the time taken to travel a distance at a constant speed - it varies directly with distance and inversely with speed.

Step by step solution

01

Understand Combined Variation

Combined variation takes place when a quantity varies directly as the product or quotient of two or more other quantities. It is a combination of direct variation and inverse variation. For example, if y varies directly as x and inversely as z, then the equation can be represented as \(y = k * x / z\). Here, \(k\) is the constant of variation.
02

Direct and Inverse Variation

In direct variation, as one quantity increases, the other quantity also increases. Similarly, as one quantity decreases, the other quantity also decreases. In contrast, in inverse variation, as one quantity increases, the other quantity decreases and vice versa.
03

Example of Combined Variation

Consider a simple situation where a car travels at a constant speed. The time taken (t) to travel a certain distance (d) varies directly as the distance and inversely as the speed (s) of the car. This can be written as \(t = k * d / s\). So, if the distance doubles (keeping speed constant), the time taken will also double. Similarly, if the speed doubles (keeping distance constant), the time taken will be halved.

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Key Concepts

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

Direct Variation
Direct variation implies a relationship between two quantities where they move in tandem. When one quantity increases or decreases, the other does so proportionally. This means if you double one quantity, you also double the other. The mathematical expression often looks like this: \( y = kx \). Here, \( y \) varies directly with \( x \), and \( k \) is the constant of variation. It dictates the rate at which \( y \) and \( x \) relate to each other.
  • Example: Suppose the cost \( C \) of buying apples is directly proportional to the number of apples \( n \). If each apple costs $3, then \( C = 3n \). If you buy twice as many apples, the total cost doubles.
  • Key Point: The graph of a direct variation is a straight line through the origin.
Inverse Variation
Inverse variation is like a balancing act between two quantities. Here, when one quantity goes up, the other goes down. It is the opposite of direct variation. You can write the relationship as \( y = \frac{k}{x} \). This indicates that \( y \) varies inversely with \( x \), and again, \( k \) is the constant of variation.
  • Example: Imagine the speed \( s \) of a vehicle is inversely proportional to the time \( t \) taken to travel a fixed distance. If the speed is reduced, it takes longer to travel the same distance. Mathematically, this is \( t = \frac{k}{s} \).
  • Key Point: The graph of an inverse variation is a hyperbola.
Constant of Variation
The constant of variation, denoted as \( k \), serves as a bridge that maintains the relationship in both direct and inverse variations. For direct variations, it represents a fixed ratio between two variables, ensuring that as one increases or decreases, the other does so proportionally. On the other hand, in inverse variations, \( k \) keeps the product of the related variables constant.
  • Role in Equations: In \( y = kx \), \( k \) is the constant multiplier, while in \( y = \frac{k}{x} \), it is the product of \( y \) and \( x \).
  • Important Factor: It remains unchanged during the variation process, signifying the proportional relationship between the variables involved.
  • Units: The constant typically carries units that ensure consistent results across calculated variables.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form quotient \(+\frac{\text { remainder }}{\text { divisor }}\) Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g\) $$g(x)=\frac{3 x-7}{x-2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \((x+3)^{2}, x \neq-3,\) resulting in the equivalent inequality \((x-2)(x+3)<2(x+3)^{2}\)

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

Will help you prepare for the material covered in the next section. Rewrite \(4-5 x-x^{2}+6 x^{3}\) in descending powers of \(x\)
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