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Chapter 2: Problem 48

Describe in words the variation given by the equation. $$Q=\frac{k p^{2}}{q^{3}}$$

Short Answer

Expert verified
Q is directly proportional to the square of p and inversely proportional to the cube of q.

Step by step solution

01

Identify the Variables

The equation given is written in terms of the variables \(Q\), \(p\), \(q\), and \(k\). These variables all potentially represent different quantities, and \(k\) is often a constant.
02

Determine How \(Q\) Varies with \(p\)

Focus on the numerator, \(p^{2}\). Since \(p\) is squared, as \(p\) increases or decreases, \(Q\) will vary with the square of \(p\). Thus, \(Q\) is directly proportional to the square of \(p\).
03

Determine How \(Q\) Varies with \(q\)

Next, look at the denominator, \(q^{3}\). As \(q\) increases or decreases, \(Q\) will vary with the cube of \(q\). Hence, \(Q\) is inversely proportional to the cube of \(q\).
04

Combine the Variations

Since \(Q\) is directly proportional to \(p^{2}\) and inversely proportional to \(q^{3}\), the overall variation is described by the equation. This means that changes in \(p\) and \(q\) affect \(Q\) according to these relations.
05

Consider the Constant \(k\)

The constant \(k\) typically scales the relationship but does not affect how \(Q\) varies with \(p\) and \(q\). It's important to keep in mind that \(k\) stays unchanged when considering the variation.

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Key Concepts

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

proportional relationships
A proportional relationship occurs when two quantities increase or decrease at the same rate. If one quantity is always a constant times another, they have a proportional relationship.
This relationship can be represented by the formula: \(y = kx\), where \(k\) is a constant. When \(y\) is directly proportional to \(x\), doubling \(x\) would double \(y\).
Similarly, if \(k\) is negative, it indicates an inverse proportionality. Understanding these relationships helps in solving various mathematical and real-world problems.
direct variation
Direct variation describes a relationship between two variables where one is a constant multiple of the other. In simpler terms, if one variable changes, the other changes in the same direction. For example, if \(y\) directly varies with \(x\), then: \(y = kx\), where \(k\) is the constant of proportionality.
  • If \(k\) is positive, both variables increase or decrease together.
  • If \(k\) is negative, one variable increases while the other decreases.
In our exercise, \(Q\) varies directly with \(p^2\), meaning if \(p\) increases, \(Q\) increases by the square of \(p\).
Recognizing direct variation helps in understanding how quantities relate and scale with each other, which is crucial for accurate modeling and predictions.
inverse variation
Inverse variation is where one variable increases as the other decreases. When two variables are inversely proportional, their product is constant, expressed by: \(xy = k\) or \(y = \frac{k}{x}\).
In our given exercise, \(Q\) varies inversely with \(q^3\), meaning as \(q\) increases, \(Q\) decreases by the cube of \(q\).
To understand inverse variation better, consider:
  • If \(q\) is doubled, \(Q\) is reduced to one-eighth of its original value, since \(q^{3}\text{ increases by a factor of 8}\).
  • If \(q\) is halved, \(Q\) increases by a factor of 8.
This concept plays a vital role in understanding phenomena where one element diminishes as another grows, like gravitational force diminishing with distance.

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Most popular questions from this chapter

Determine algebraically whether the function is even, odd, or neither even nor odd. Then check your work graphically, where possible, using a graphing calculator. $$f(x)=\sqrt{x^{2}+1}$$

Determine whether the statement is true or false. The product of an even function and an odd function is odd.

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