Point on a parabola where the tangent intersects a point

[sp09ta]

1. A car is traveling on a highway shaped like a parabola with its vertex at the origin. The car begins at a point 100m west and 100m north of the origin and is traveling easterly. There is a statue 100m east and 50m north of the origin. At what point on the highway will the car`s headlights illuminate the statue.
2. d/dx x^n = x*x^(n-1)
(y2-y1)/(x2-x1)=m

3. The car begins at (-100,100) and travels on the domain [-100,inf).

let f(x) represent the path of the car, thus f`(x) will represent the car`s headlights.

f(x)=(1/100)*x^2 and f'(x)=(1/50)*x

We need to find point P, who's tangent line intersects (100,50).

Since the equation of the tangent is y=mx+b, and m=f'(x), then (50-y)/(100-x)=(1/50)x

I then solved for y to receive y=(x^2+100x+2500)/50

and for y=0, x=50.

So sub y=50 into my original eqn:

Thus, 50=(1/100)*x^2 so, 50sqrt(2)=x

And the statue will be illuminated by the car`s headlights when the car is at (50*sqrt(2), 50)?


Does this look correct?

 

[mighty2000]

Yes, your solution looks correct. You have correctly used the tangent line equation to find the point of intersection between the tangent line and the statue's location. Your calculations and reasoning are also sound. Good job!