Precalc Variation Help

Question

Suppose z varies directly as x and inversely as the square of y. If x is doubled and y is tripled, what happens to z?

Answer 1

z = kx / y²

If x is doubled and y is tripled, then we have k(2x) / (3y)² = (2/9)(kx / y²) = (2/9)z

The "new" z value is 2/9 of the "old" value of z.

Answer 2

First, let's review variation definitions:

Direct variation describes one variable as a constant multiple of another variable.

Inverse (Indirect variation) describes one variable increasing and the other one decreasing.

So let's express what the problem supposes:

(The equal sign below expresses the assumption of a proportional relationship.)

z = x

z = 1/y^2

Next, we'll "double" x.

z = [2] * x (algebra: once done to one side is done to the other) --> 2z = 2x

And, we'll triple y.

z = 1/([3]y^2) --> 1/3 z = z/3 = 1/(3y^2)

So what happened to z is that it was multiplied by 2 due to the doubling of x in the first expression and it was divided by 3 due to the tripling of y in the second expression (more simply, z is now multiplied by 2/3):

z --> 2/3 z = 2z/3

Hope this helps and Good Luck!

-Michael Mount