Joint and combined variation

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Intros

Lessons

Introduction to joint and combined variation
Review: direct variation vs. inverse variation
What is a joint variation?
What is a combined variation and how is it different from a joint variation?
How to solve a variation problem?

Examples

Lessons

Identifying Types of Variations
Determine whether each equation represents a direct, inverse, joint, or combined variation. Name the constant of variation.
1. $xy = 17$
2. $p = 5q$
3. $b = \frac{3ac}{4}$
4. $m = \frac{n}{8}$
5. $e = \frac{5f}{7g}$
Translating Variation Statements Into Equations
Translate the following statements, and then classify the variations.
1. $x$ varies jointly as $y$ and the square of $z$ .
2. The speed of a race car varies directly as the distance and inversely as the time.
Solving Variation Problems
Find the missing variables.
1. $a$ varies directly with $b$ and $c$ . If $a=336$ when $b=4$ and $c=7$ , find $a$ when $b=2$ and $c=11$ .
2. $p$ 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.
1. Find the constant of variation $k$ . Round your answer to 2 decimal places.
2. What is the volume of a can that has a 7 cm height and 3 cm radius?
The time required to process a shipment at Mamazon varies directly with the number of orders being made and inversely with the number of workers. If 1344 orders can be processed by 7 workers in 12 hours, how long will it take 125 workers to process 20,000 items?

Topic 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.

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