find the point (x,y) on the graph of the equation y=2√x which is closest to the point (11,0). use this to write down the minimum distance from the graph to the point (11,0.)

Question

find the point (x,y) on the graph of the equation y=2√x which is closest to the point (11,0).  use this to write down the minimum distance from the graph to the point (11,0.)

dumbcow · Answer

use distance equation
$$d^2 = (x-11)^2 + (2 \sqrt{x} - 0)^2$$
$$d^2 = x^2 -18x + 121$$
differentiate
$$2d d' = 2x - 18$$
$$d' = \frac{2x - 18}{2d} = \frac{x-9}{d}$$
minimize distance by setting derivative equal to zero
$$\frac{x-9}{d} = 0  \rightarrow x = 9$$
point closest to (11,0) is at x=9
$$(9,6)$$
min distance is:
$$\sqrt{40} = 2\sqrt{10}$$