Let S denote the set of points with coordinates (x,y) such that 0 < or = x < or = 15 & y^2= 150-10x?

Question

Find the coordinates of the point(s) of s nearest to the orgin. Also find the coordinates of the point(s) of S farthest from the orgin. Justify your answer.

Answer 1

You have to set up equation for distance from a point on the curve to orgin, find differinial and set to 0.
The equation is squareroot((x-0)^2+(y-0)^2),or by substution, squareroot((x-0)^2+(( squareroot(150-10x))-0)^2).
Or, sqr(x^2+150-10x).

Differinate it, and setting to zero gives you, (2x-10)/(sqr(x^2+150-10x)) = 0.

SO, 2x-10=0, so x=5.
At point (5,10) its at the closest.
Now, to find furthest, look at boundaries, x=0, and x=15.
At x=15, distance is 15.
AT x=0, distance is sqr(150) = 12.247 approximately.
SO, farthest is at (15,0).

Answer 2

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