Section 2.6: Variation and Applications – Precalculus Study Notes

Variation and Applications

Objectives

This section introduces the concepts of direct, inverse, and combined variation, and demonstrates how to find equations of variation given values of the variables. It also covers solving applied problems involving these types of variation.

• Find equations of direct, inverse, and combined variation given variable values.

• Solve applied problems involving variation.

Direct Variation

Definition and Properties

Direct variation describes a relationship where one variable is a constant multiple of another. This gives rise to a linear function of the form $f(x) = kx$ or $y = kx$, where k is a positive constant called the variation constant or constant of proportionality.

• Direct variation: $y = kx$

• y varies directly as x: y increases as x increases.

• Graph: The graph of $y = kx$ (for $k > 0$) passes through the origin and rises from left to right.

• Slope: The constant $k$ is the slope of the line.

Example: Finding the Variation Constant

Suppose $y$ varies directly as $x$, and $y = 32$ when $x = 2$.

• Equation: $y = kx$

• Substitute values: $32 = k imes 2$

• Solve for $k$: $k = \frac{32}{2} = 16$

• Variation constant is 16. The equation of variation is $y = 16x$.

Example: Applied Problem – Melting Snow

The number of centimeters of water $W$ produced from melting snow varies directly as the number of centimeters of snow $S$. If 150 cm of snow yields 16.8 cm of water, how much water will 200 cm of snow produce?

• Equation: $W(S) = kS$

• Find $k$: $W(150) = k \times 150 = 16.8 \implies k = \frac{16.8}{150} = 0.112$

• Equation of variation: $W(S) = 0.112S$

• Find $W(200)$: $W(200) = 0.112 \times 200 = 22.4$

• Result: 200 cm of snow will melt to 22.4 cm of water.

Inverse Variation

Definition and Properties

Inverse variation describes a relationship where one variable is a constant divided by another. The function is of the form $f(x) = \frac{k}{x}$ or $y = \frac{k}{x}$, where k is a positive constant.

• Inverse variation: $y = \frac{k}{x}$

• y varies inversely as x: y decreases as x increases.

• Graph: The graph of $y = \frac{k}{x}$ (for $k > 0$) is a hyperbola, decreasing on the interval $(0, \infty)$.

• Constant of proportionality: k

Example: Finding the Variation Constant

Suppose $y$ varies inversely as $x$, and $y = 16$ when $x = 0.3$.

• Equation: $y = \frac{k}{x}$

• Substitute values: $16 = \frac{k}{0.3}$

• Solve for $k$: $k = 16 \times 0.3 = 4.8$

• Variation constant is 4.8. The equation of variation is $y = \frac{4.8}{x}$.

Example: Applied Problem – Filling a Swimming Pool

The time $t$ required to fill a swimming pool varies inversely as the rate of flow $r$ of water into the pool. If a tank truck can fill a pool in 90 min at a rate of 1500 L/min, how long would it take at a rate of 1800 L/min?

• Equation: $t(r) = \frac{k}{r}$

• Find $k$: $t(1500) = \frac{k}{1500} = 90 \implies k = 90 \times 1500 = 135,000$

• Equation of variation: $t(r) = \frac{135,000}{r}$

• Find $t(1800)$: $t(1800) = \frac{135,000}{1800} = 75$

• Result: It would take 75 min to fill the pool at a rate of 1800 L/min.

Solving Variation Problems

General Steps

To solve variation problems, follow these steps:

1. Determine whether direct or inverse variation applies.

2. Write an equation of the form $y = kx$ (direct) or $y = \frac{k}{x}$ (inverse), substitute known values, and solve for $k$.

3. Write the equation of variation and use it to find unknown values.

Combined Variation

Types and Equations

Combined variation involves more complex relationships, including powers and multiple variables.

• Direct variation as nth power: $y = kx^n$

• Inverse variation as nth power: $y = \frac{k}{x^n}$

• Joint variation: $y = kxz$ (y varies jointly as x and z)

Example: Direct Variation as a Power

Suppose $y$ varies directly as the square of $x$, and $y = 12$ when $x = 2$.

• Equation: $y = kx^2$

• Substitute values: $12 = k \times 2^2 = k \times 4$

• Solve for $k$: $k = \frac{12}{4} = 3$

• Equation of variation: $y = 3x^2$

Example: Joint and Inverse Variation

Suppose $y$ varies jointly as $x$ and $z$, and inversely as the square of $w$, with $y = 105$ when $x = 3$, $z = 20$, and $w = 2$.

• Equation: $y = k \frac{xz}{w^2}$

• Substitute values: $105 = k \frac{3 \times 20}{2^2} = k \frac{60}{4} = k \times 15$

• Solve for $k$: $k = \frac{105}{15} = 7$

• Equation of variation: $y = 7 \frac{xz}{w^2}$

Example: Volume of a Tree

The volume $V$ of wood in a tree varies jointly as the height $h$ and the square of the girth $g$. If $V = 216$ m3 when $h = 30$ m and $g = 1.5$ m, what is the height of a tree whose volume is $344$m3 and girth is $1.6$ m?

• Equation: $V = k h g^2$

• Find $k$: $216 = k \times 30 \times (1.5)^2 = k \times 30 \times 2.25 = k \times 67.5$

• Solve for $k$: $k = \frac{216}{67.5} = 3.2$

• Equation: $V = 3.2 h g^2$

• Substitute new values: $344 = 3.2 \times h \times (1.6)^2 = 3.2 \times h \times 2.56 = 8.192 h$

• Solve for $h$: $h = \frac{344}{8.192} \approx 42$

• Result: The height of the tree is about 42 m.

Summary Table: Types of Variation

Type	General Equation	Relationship
Direct Variation	$y = kx$	y increases as x increases
Inverse Variation	$y = \frac{k}{x}$	y decreases as x increases
Direct Variation (nth power)	$y = kx^n$	y increases as x increases (nonlinear)
Inverse Variation (nth power)	$y = \frac{k}{x^n}$	y decreases as x increases (nonlinear)
Joint Variation	$y = kxz$	y varies as product of x and z
Combined Variation	$y = k \frac{xz}{w^n}$	y varies jointly as x and z, inversely as nth power of w

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