Joint and combined variation

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  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?
  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?