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)