Solving Calculus Question: x^4 + y^4 = 17, x + 8y = 30

[orange]

Hey hey.

I have this question I have a hard time figuring out. Here is the assignment as I got it:

What points are the closests between the functions

x^4 + y^4 = 17

and

x + 8y = 30?

I have started a few times but never really got far. I thought the line between the closets points must be a right angle with the plane x + 8y = 30, which might be solved with dot products. That, and that x^4 + y^4 = 17 describes some type of circle, but I can't really see it in front of me.

Anyone have any pointers? :!)

 

[finchie_88]

Well, one way of doing it would be to find dy/dx for both, and then make the dy/dx for the first equation equal the dy/dx for the second. You should get a function in x and y, then solve simultaniously with either of the equations of the lines. Like I said, not the best way, but it should work. You could try and do it with cross products if you like.

e.g.
$4x^3 + 4y^3 \frac{dy}{dx} = 0$ then do the same for the other one and substitute one into the other.

 

[Fermat]

x^4 + y^4 = 17 is like a square with curved corners (centred on the origin).
x + 8y = 30 isn't a plane, it's a straight line.

What you have to do is find a point (Px, Py) on the x^4 + y^4 = 17 curve where the slope is the same as the slope of the straight line.
If you draw this tangent, it will be equidistant everywhere to the straight line x + 8y = 30.
Now all you have to do is find the distance beteen a point P and a straight line, x + 8y = 30.