how to change the order of x, y ,z?

Question

if we have <0<y<1 , 0<z<(1-y)^2 , y^2+z^2<x<1

how can we change the order of x,y,z and write 0<x<1 , 0<y<x^1/2 , 0<z<(x-y^2)

Answer 1

Hi Dorsa,

I think the easiest way to change the "old" x,y,z ranges to the "new" ranges is to first compare them. We'll end up with 3 algebraic equations that we have to solve.

Let's look at x, y, and z ranges, each individually.

For x...

We have y²+z²<x<1 (old)

and we want 0<x<1 (new)

The maximum for each range is already 1. So we just need to worry about the minimum.

In order for the old range to match the new range, we see that y²+z²=0. This is our first algebraic equation. But, we can't solve it without an equation for y and z.

For y...

We have 0<y<1

and we want 0<y<x^1/2

Again, let's compare the max's and min's for each. Both have a minimum of 0. In order for the maxima to be equal, 1=x^1/2. This is our second algebraic equation.

For z...

We have 0<z<(1-y)²

and we want 0<z<(x-y²)

Let's compare the max's and min's again. This time, we get (1-y)²=x-y².

Now, solve your 3 algebraic equations to find x,y,z..

y²+z² = 0

(1-y)²= x-y²

1 = x^1/2

See if you can solve for x, y, z yourself. Hope that helps (: