1.Locate the points on the curve y=x^2 that are closest to the point (0,4.5)?

2.A rectangle is inscribed in a circle of radius 5m.Determine the maximum possible area for such a rectangle?

Please provide full steps!(Doh)

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- Apr 18th 2008, 08:09 PMsomeone21Functional Grpahs and optimization
1.Locate the points on the curve y=x^2 that are closest to the point (0,4.5)?

2.A rectangle is inscribed in a circle of radius 5m.Determine the maximum possible area for such a rectangle?

Please provide full steps!(Doh) - Apr 19th 2008, 12:00 AMOpalg
We want to minimise a distance, but in practice it's easier to minimise the square of the distance.

The distance d from a point (x,y) to the point (0,4.5) is given by $\displaystyle d^2 = (x-0)^2 + (y-4.5)^2$. If the point (x,y) lies on the curve y=x^2 then we can replace x^2 by y, which gives $\displaystyle d^2 = y + (y-4.5)^2$. Multiply out, complete the square, and you'll find the value of y that minimises d^2.