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Chapter 3: Problem 44

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

Short Answer

Expert verified
Joint variation is a situation in which a quantity varies directly as the product of two or more other quantities. For example, if \(y = kxz\) where \(k\) is the constant of variation, then \(y\) varies jointly as \(x\) and \(z\).

Step by step solution

01

Understanding Joint Variation

Joint variation is a situation in which a quantity varies directly as the product of two or more other quantities. It is expressed in the form \(y = kxz\), where \(y\) varies jointly as \(x\) and \(z\), and \(k\) is the constant of variation, which remains the same throughout.
02

Example of Joint Variation

For example, let's suppose that \(y\) varies jointly as \(x\) and \(z\). If \(x=2\), \(z=3\), and \(y=12\), we can figure out the constant of variation \(k\) by rearranging the equation to \(k=y/(xz)\) and substituting. On doing so, \(k=12/(2*3)\) which gives \(k=2\). We can substitute \(k\) back into the equation to verify that \(y = kxz\). So \(y = 2*2*3\) which equals to \(12\). Therefore, in this example, y varies jointly as \(x\) and \(z\) with \(k\) equal to \(2\).
03

Important points to note

Key points to note about joint variation are that the constant of variation \(k\) remains the same throughout, and that the variable \(y\) varies as the product of \(x\) and \(z\), not as their sum or difference.

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Most popular questions from this chapter

Use the four-step procedure for solving variation problems given on page 424 to solve Exercises 1-10. \(y\) varies directly as \(x . y=65\) when \(x=5 .\) Find \(y\) when \(x=12 .\)

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x\) and inversely as the square of \(z . y=20\) when \(x=50\) and \(z=5 .\) Find \(y\) when \(x=3\) and \(z=6\).

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