Find the point p on the line y=3x that is closest to the point (30,0). what is the least distance between p and (30,0)?

P will be the point of intersection between the given line and a perpendicular line through the given point.

The given line has a slope (x-coefficient) of 3, so the perpendicular line will have a slope that is the negative reciprocal of that: -1/3. In point-slope form, the line through the given point is then ...

... y = (-1/3)(x -30)

The point of intersection can be found by setting the y-values equal.

... 3x = y = (-1/3)(x - 30)

... 9x = -x +30 . . . . . . . . . . multiply by 3, eliminate parentheses

... 10x = 30 . . . . . . . . . . . . add x

... x = 3 . . . . . . . . . . . . . . . . divide by 10

Now, either equation can be used to find y. It is convenient to use the simpler equation:

... y = 3x = 3·3 = 9

So, the point of intersection is ...

... P = (3, 9)

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The distance between this point and the given point can be found using the Pythagorean theorem.

The straight-line distance between the points is ...

... d = √(27² +9²) = 9√(3² +1²) = 9√10