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What does "varies directly" mean?

Could someone please expalin to me what it means when two variables "vary directly", and help me solve the following problem?

If "a" varies directly with "b" squared, and if b=4 and a=3, what is the value of "a" when b=8?

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Best Answer:
it means x=ky. or in this case a=kb. k being the constant of variation.
so 3=k4. a=3 when b=4. so divide both sides to find the constant of
variation, k. you get k=3/4. so a=3/4b. if b=8, a=3/4(8).

a=6, when b=8

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Equations can vary either directly or inversely.

If y varies directly with x, for example, a typical equation would look
like this: y = 5x (with "5" being a placeholder for your constant). As x
increases, y increases, hence they vary DIRECTLY.

If y varies inversely with x, you may see something like: y = 5/x. In
this equation, as x increases, y DECREASES (which is considered an
inverse relationship).

So, in your problem, the basic equation is: a = k(b^2) ***The k here just represents the unknown constant***

a = k(b^2)

Plug in 4 for b and 3 for a, as per the first part of the problem.

3 = k(16)

3 = 16k

k = 3/16

So, now we take the value that we found for k and use it to solve the second part of the problem.

a = 3/16 (b^2)

plug in 8 for b

a = 3/16 (64)

a = 12

Ta-da! If you have any more questions, feel free to ask.

EDIT: Pikajew6, I think you may have glossed over the beginning of the
problem where the poster said, " "a" varies directly with "b"
squared...".