The quantity y varies directly with the square of x and inversely with z. When x is 9 and z is 27, y is 6. What is the constant of variation?

Answer 1

The constant of variation is #k=2#.

To say that a variable "varies directly" with some quantity, we mean that the variable scales with that quantity. In this example, that means the scaling of #y# is "in sync" with the scaling of #x^2# (i.e. when #x^2# doubles, #y# also doubles).

We're also given that #y# varies inversely with #z#, meaning that when #z# doubles, #y# gets halved.

We can take the information given and form it into a single equation like this:

#y=kx^2/z#

The #k# is the constant of variation we seek. Plugging in the given values of #x#, #y#, and #z# into this equation, we get

#6=k*(9^2)/(27)#

#6=k*81/27#

#6=k*3#

#2=k#

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Answer 2

Nathan Axel

The constant of variation is ( \frac{y \cdot z}{x^2} ), which in this case is ( \frac{6 \cdot 27}{9^2} = 2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.