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Joint and combined variation

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Intros
Lessons
  1. Introduction to joint and combined variation
  2. Review: direct variation vs. inverse variation
  3. What is a joint variation?
  4. What is a combined variation and how is it different from a joint variation?
  5. How to solve a variation problem?
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Examples
Lessons
  1. Identifying Types of Variations
    Determine whether each equation represents a direct, inverse, joint, or combined variation. Name the constant of variation.
    1. xy=17xy = 17
    2. p=5qp = 5q
    3. b=3ac4b = \frac{3ac}{4}
    4. m=n8m = \frac{n}{8}
    5. e=5f7g e = \frac{5f}{7g}
  2. Translating Variation Statements Into Equations
    Translate the following statements, and then classify the variations.
    1. xx varies jointly as yy and the square of zz.
    2. The speed of a race car varies directly as the distance and inversely as the time.
  3. Solving Variation Problems
    Find the missing variables.
    1. aa varies directly with bb and cc. If a=336a=336 when b=4b=4 and c=7c=7, find aa when b=2b=2 and c=11c=11.
    2. pp varies directly as qq but inversely as rr. If p=14p=14 when q=2q=2 and r=5r=5, find qq when p=105p=105 and r=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.
    1. Find the constant of variation kk. 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?
  5. 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
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    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⋅x⋅zy=k \cdot x \cdot z where kk is the constant of variation and k≠0k \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. yy varies directly with xx and inversely with zz (y=k⋅xz) (y = k \cdot \frac{x}{z})
    • Steps to solving a variation problem:
      1. Write the general variation formula of the problem.
      2. Find the constant of variation kk.
      3. Rewrite the formula with the value of kk.
      4. Solve the problem by inputting known information.