Question

Suppose that y varies inversely with the square root of x and y=50 when x=4, how do you find y when x=5?

Answer 1

If `y` varies inversely with `sqrt(x)`
then
`y*sqrt(x) = c` for some constant `c`

Given `(x,y)=(4,50)` is a solution to this inverse variation
then
`50*sqrt(4) = c`
`rarr c = 100 color(white)("xxxxxxxxxx")` (see note below)
and the inverse variation equation is
`y*sqrt(x) = 100`

When `x = 5` this becomes
`y*sqrt(5) = 100`

`sqrt(5) = 100/y`

`5 = 10^4/y^2`

`y = sqrt(5000) = 50sqrt(2)`

Note: I have interpreted "y varies inversely with the square root of x" to mean the positive square root of x (i.e. `sqrt(x)`) which also implies that y is positive. If this is not the intended case, the negative version of y would also need to be allowed.