1. ## Optimization Problem

Hi, I had a problem that I just had no idea how to do. I was told it was an optimization but I am still confused. The problem states :
Let S denote the set of points with the coordinates (x,y) such that 0≤x≤15 and y^2 = 150- 10x. Find the coordinates of the point(s) of S nearest to the origin. Also find the coordinates of the point(s) of S farthest from the origin. Justify your answer.
All i have gotten is y= squareroot(150-10x) and that is not much.

2. Originally Posted by calc_help123
Hi, I had a problem that I just had no idea how to do. I was told it was an optimization but I am still confused. The problem states :
Let S denote the set of points with the coordinates (x,y) such that 0≤x≤15 and y^2 = 150- 10x. Find the coordinates of the point(s) of S nearest to the origin. Also find the coordinates of the point(s) of S farthest from the origin. Justify your answer.
All i have gotten is y= squareroot(150-10x) and that is not much.
remember the distance formula ? distance of the point (x,y) from the origin is ...

$\displaystyle d = \sqrt{(x - 0)^2 + (y - 0)^2}$

$\displaystyle d = \sqrt{x^2 + y^2}$

substitute for $\displaystyle y^2$

$\displaystyle d = \sqrt{x^2 - 10x + 150}$

let $\displaystyle d = \sqrt{z}$

if z is maximized/minimized, so is d ...

$\displaystyle z = x^2 - 10x + 150$

$\displaystyle \frac{dz}{dx} = 2x - 10$

$\displaystyle 2x - 10 = 0$

$\displaystyle x = 5$

$\displaystyle \frac{d^2z}{dz^2} = 2 > 0$

$\displaystyle x = 5$ will yield a minimum distance

the endpoints, $\displaystyle x = 0$ and $\displaystyle x = 15$ , will be maximums ... calculate which yields the absolute maximum distance.