If Z varies directly with x and inversely with y^2 when x=2 and y=5, z=8 what is the value of z when x=4 and y=9?

May 24, 2016

$z = 4.9383$

Explanation:

Direct Variation is $y = k x$
Inverse Variation is $y = \frac{k}{y}$

If $z$ varies directly to $x$ the equation would be $z = k x$

If $z$ varies inversely to ${y}^{2}$ the equation would be $z = \frac{k}{y} ^ 2$

Combining these two equations we get $z = \frac{k x}{y} ^ 2$

We use the first set of values to solve for the constant of variation $k$

$x = 2$
$y = 5$
$z = 8$

$8 = \frac{2 k}{5} ^ 2$

$8 = \frac{2 k}{25}$

$8 \left(25\right) = \frac{2 k}{\cancel{25}} \cancel{25}$

$200 = \left(2 k\right)$

$\frac{200}{2} = \frac{\cancel{2} k}{\cancel{2}}$

$100 = k$

Now use the constant $k$ with the second values to solve for $z$

$x = 4$
$y = 9$
z=?
$k = 100$

$z = \frac{\left(100\right) \left(4\right)}{9} ^ 2$

$z = \frac{400}{81}$

$z = 4.9383$