Question

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`.

Explanation:

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`