1.11 Modeling Variation

1.11.1 Learning Objectives

By the end of this section, you should be able to:

  • recognize direct, inverse, joint, and combined variation;
  • translate variation statements into equations;
  • determine constants of proportionality; and
  • make predictions using variation models.

1.11.2 Direct Variation

Variation models describe proportional relationships.

Direct variation has the form

\[ y=kx. \]

The ratio \(y/x\) is constant, and the graph is a line through the origin.

Example 1.54 Direct Variation

Suppose \(y\) varies directly as \(x\), and \(y=18\) when \(x=6\).

\[ 18=6k\Longrightarrow k=3. \]

The model is \(\boxed{y=3x}\). When \(x=11\), \(y=33\).

Exercise 1.59 If \(y\) varies directly as \(x\) and \(y=20\) when \(x=4\), find \(k\).

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Answer: \(5\)

Exercise 1.60 If \(y=7x\), find \(y\) when \(x=9\).

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Answer: \(63\)

1.11.3 Inverse Variation

Inverse variation has the form

\[ y=\frac{k}{x},\qquad x\ne0. \]

The product \(xy\) is constant. Increasing one positive variable decreases the other.

Example 1.55 Inverse Variation

Suppose \(p\) varies inversely as \(v\), and \(p=12\) when \(v=5\).

\[ 12=\frac{k}{5}\Longrightarrow k=60. \]

Thus \(\boxed{p=60/v}\). When \(v=8\), \(p=7.5\).

Exercise 1.61 If \(y\) varies inversely as \(x\) and \(y=6\) when \(x=5\), find the model.

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Answer: \(y=30/x\)

1.11.4 Joint Variation

Joint variation has the form

\[ z=kxy, \]

so \(z\) varies directly with both \(x\) and \(y\).

Example 1.56 Joint Variation

Suppose \(z\) varies jointly as \(x\) and \(y\), and \(z=48\) when \(x=4\) and \(y=3\).

\[ 48=k(4)(3)\Longrightarrow k=4. \]

The model is \(\boxed{z=4xy}\).

Exercise 1.62 If \(z=2xy\), find \(z\) when \(x=3\) and \(y=8\).

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Answer: \(48\)

1.11.5 Combined Variation

Variation may also be combined. For example,

\[ z=\frac{kx}{y^2} \]

means that \(z\) varies directly with \(x\) and inversely with the square of \(y\).

Example 1.57 Combined Variation

Suppose \(w\) varies directly as \(x\) and inversely as \(y^2\). If \(w=10\) when \(x=8\) and \(y=2\), then

\[ 10=\frac{k(8)}{2^2}\Longrightarrow k=5. \]

Therefore,

\[ \boxed{w=\frac{5x}{y^2}}. \]

Exercise 1.63 Translate: \(P\) varies directly as \(m\) and inversely as \(r^2\).

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Answer: \(P=km/r^2\)

1.11.6 Building a Variation Model

To build a model:

  1. Translate the variation statement into an equation with \(k\).
  2. Substitute known values to determine \(k\).
  3. Write the completed model.
  4. Use it to find the requested quantity.

1.11.7 Conceptual Takeaways

  • Direct variation keeps a ratio constant.
  • Inverse variation keeps a product constant.
  • Joint variation involves direct proportionality to two or more variables.
  • The constant \(k\) is determined from known data.
  • Verbal phrases such as “as,” “jointly,” and “inversely as the square” determine the model’s structure.

1.11.8 Skills You Should Be Able to Do

  • Identify types of variation from equations or descriptions.
  • Write a variation equation with a proportionality constant.
  • Determine \(k\) from data.
  • Use a completed model to make predictions.
  • Explain how changing one variable affects another.

1.11.9 Practice Problems with Solutions

  1. \(y\) varies directly as \(x\). If \(y=35\) when \(x=7\), find \(y\) when \(x=12\).

    Show Solution

    \[ 35=7k\Longrightarrow k=5. \] Thus \(y=5x\), and when \(x=12\), \[ \boxed{y=60}. \]

  2. \(y\) varies inversely as \(x\). If \(y=9\) when \(x=4\), find \(x\) when \(y=6\).

    Show Solution

    \[ 9=\frac{k}{4}\Longrightarrow k=36. \] Then \[ 6=\frac{36}{x}\Longrightarrow\boxed{x=6}. \]

  3. \(z\) varies jointly as \(x\) and \(y\). If \(z=30\) when \(x=3\) and \(y=5\), find \(z\) when \(x=8\) and \(y=2\).

    Show Solution

    \[ 30=k(3)(5)\Longrightarrow k=2. \] Therefore, \[ z=2(8)(2)=\boxed{32}. \]

  4. \(w\) varies directly as \(x^2\) and inversely as \(y\). If \(w=18\) when \(x=3\) and \(y=2\), find \(w\) when \(x=5\) and \(y=4\).

    Show Solution

    \[ 18=\frac{k(3^2)}2\Longrightarrow k=4. \] Then \[ w=\frac{4(5^2)}4=\boxed{25}. \]

  5. The time \(t\) required to complete a fixed trip varies inversely with speed \(v\). A trip takes \(4\) hours at \(75\) km/h. How long does it take at \(100\) km/h?

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    Since \(t=k/v\), \[ 4=\frac{k}{75}\Longrightarrow k=300. \] At \(100\) km/h, \[ t=\frac{300}{100}=\boxed{3\text{ hours}}. \]

  6. The area \(A\) of a circle varies directly as the square of its radius. Use a circle of radius \(2\) and area \(4\pi\) to find the variation equation.

    Show Solution

    \[ A=kr^2. \] Using \(4\pi=k(2^2)\) gives \(k=\pi\), so \[ \boxed{A=\pi r^2}. \]