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Suppose a varies directly as b, and a varies inversely as c. Find b when a = 7 and c = -8, if b = 15 when c = 2 and a = 4.

User Dacx
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1 Answer

3 votes

Answer:

b = -105

Explanation:

Normally, the direct variation equation involves multiplication and is given by:

y = kx, where

  • y varies directly as x,
  • and k is the constant of proportionality.

Furthermore, the normal inverse variation equation involves division and is given by:

y = k/x, where

  • y varies inversely as x,
  • and k is the constant of proportionality

To combine direct and inverse variation, we need to use combined variation:

  • Since a varies directly as b, normally we'd have a = kb.
  • Since a also varies inversely as c, we can represent this by simply dividing kb by c.

Step 1: Thus, the entire equation representing a varying directly as b and a varying inversely as c is given by:

a = (kb)/c

Step 2: Before we can find b, we need to find k using the info we're given. Since we're told that b = 15 when c = 2 and a = 4, we can plug these values into the combined variation equation to solve for k, the constant of proportionality

4 = (15k)/2

8 = 15k

8/15 = k

Step 3: Now we can plug in 8/15 for k, 7 for a, and -8 for c into the combined variation equation, allowing us to solve for b:

7 = (8/15b)/-8

-56 = 8/15b

-105 = b

Thus, b = -105, when a = 7 and c = -8 (and only when k = 8/15)

User Jaylin
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