Joint and combined variation - Relations and Functions

Joint and combined variation

Lessons

Notes:

In this lesson, we will learn:

Identifying Types of Variations

Translating Variation Statements Into Equations

Solving Variation Problems

Word Problems of Variations

Joint variation is a direct variation, but with two or more variables. It has the equation $y=k \cdot x \cdot z$ where $k$ is the constant of variation and $k \neq 0$ .

A combined variation is formed when we combine any of the variations together (direct, inverse and joint). In most cases, we combine direct and inverse variations to form a combined variation. i.e. $y$ varies directly with $x$ and inversely with $z$ $(y = k \cdot \frac{x}{z})$

Steps to solving a variation problem:

Write the general variation formula of the problem.

Find the constant of variation $k$ .

Rewrite the formula with the value of $k$ .

Solve the problem by inputting known information.

Intro Lesson

Introduction to joint and combined variation
1.
Identifying Types of Variations
Determine whether each equation represents a direct, inverse, joint, or combined variation. Name the constant of variation.
2.
Translating Variation Statements Into Equations
Translate the following statements, and then classify the variations.
3.
Solving Variation Problems
Find the missing variables.
4.
Word Problems of Variations
The volume of a cylinder varies jointly as the height and the square of its radius. A cylinder with an 9 cm height and 6 cm radius has a volume of 1018 cm³.

Joint and combined variation

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Lessons

Notes:

Intro Lesson

Introduction to joint and combined variation

a)

Review: direct variation vs. inverse variation

b)

What is a joint variation?

c)

What is a combined variation and how is it different from a joint variation?

d)

How to solve a variation problem?

1.

Identifying Types of Variations Determine whether each equation represents a direct, inverse, joint, or combined variation. Name the constant of variation.

a)

xy=17xy = 17xy=17

b)

p=5qp = 5qp=5q

c)

b=3ac4b = \frac{3ac}{4} b=43ac​

d)

m=n8m = \frac{n}{8} m=8n​

e)

e=5f7g e = \frac{5f}{7g} e=7g5f​

2.

Translating Variation Statements Into Equations Translate the following statements, and then classify the variations.

a)

xxx varies jointly as yyy and the square of zzz.

b)

The speed of a race car varies directly as the distance and inversely as the time.

3.

Solving Variation Problems Find the missing variables.

a)

aaa varies directly with bbb and ccc. If a=336a=336a=336 when b=4b=4b=4 and c=7c=7c=7, find aaa when b=2b=2b=2 and c=11c=11c=11.

b)

ppp varies directly as qqq but inversely as rrr. If p=14p=14p=14 when q=2q=2q=2 and r=5r=5r=5, find qqq when p=105p=105p=105 and r=18r=18r=18.

4.

Word Problems of Variations The volume of a cylinder varies jointly as the height and the square of its radius. A cylinder with an 9 cm height and 6 cm radius has a volume of 1018 cm3.

a)

Find the constant of variation kkk. Round your answer to 2 decimal places.

b)

What is the volume of a can that has a 7 cm height and 3 cm radius?

or

Identifying Types of Variations
Determine whether each equation represents a direct, inverse, joint, or combined variation. Name the constant of variation.

$xy = 17$

$p = 5q$

$b = \frac{3ac}{4}$

$m = \frac{n}{8}$

$e = \frac{5f}{7g}$

Translating Variation Statements Into Equations
Translate the following statements, and then classify the variations.

$x$ varies jointly as $y$ and the square of $z$ .

Solving Variation Problems
Find the missing variables.

$a$ varies directly with $b$ and $c$ . If $a=336$ when $b=4$ and $c=7$ , find $a$ when $b=2$ and $c=11$ .

$p$ varies directly as $q$ but inversely as $r$ . If $p=14$ when $q=2$ and $r=5$ , find $q$ when $p=105$ and $r=18$ .

Word Problems of Variations
The volume of a cylinder varies jointly as the height and the square of its radius. A cylinder with an 9 cm height and 6 cm radius has a volume of 1018 cm³.

Find the constant of variation $k$ . Round your answer to 2 decimal places.