Question

If z varies inversely as w, and z=10 when w=1/2, how do you find z when w=10?

Answer 1

`z=5/w" "->" at w=10 " z=1/2`

Explanation:

`color(green)("Building the equation")`

The mathematical way to show this relationship is

`z color(white)(.) alpha color(white)(.) 1/w`

This is stating that they are related but you have not yet declared the constant of variation ( conversion constant).

Let the constant of variation be `k` then we have

`" "z=kxx1/w = k/w" "->" "z=k/w`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("Determine the value of the conversion constant")`

All we now need to do is find the value of the constant `k`. This is achieved by substituting in known values.

We are told that when z=10 the value of w is `1/2`

So by substitution we have:

`" "color(brown)(z=k/w)color(blue)(" "->" "10=(color(white)(..)kcolor(white)(..))/(1/2)`

Multiply both side by `1/2` and we have:

`" "10xx1/2=kxx (color(white)(..)1/2color(white)(..))/(1/2)`

But `(color(white)(..)1/2color(white)(..))/(1/2) = 1" giving"`

`" "5=kxx1" "->" "k=5`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(green)("The final equation")`

`z=5/w`

At `w=10` we have `z=5/10 =1/2`