Car is traveling along a parabola with vertex at origin. Car starts at point 100m west and 100m north?
of the origin and travels in a easterly direction. There is a statue at 100m east and 50 m north of the origin. (statue is not on the highway) At what point on the highway will the car's headlights illuminate the statue. Express answer as ordered pair.
I know I need to find the derivative at which the line intersects the statue.
- DWReadLv 71 decade agoFavorite Answer
The start point is NW of the vertex; I'm assuming the parabola opens upwards because then the car continues east throughout. The general equation for an up-opening parabola with vertex at the origin is
y = ax²
(-100, 100) is a point on the parabola, so a=0.01 and the equation becomes
y = 0.01x²
The car's headlights are tangent to the parabola. The slope of the tangent is
y' = 0.02x
The statue is at (100, 50). Let (x₁, 0.01x₁²) be the coordinates of the car when its headlights hit the statue.
The slope of the parabola at this point is 0.02x₁.
The slope of the line joining the car and statue is (50-0.01x₁²)/(100-x₁).
(50-0.01x₁²)/(100-x₁) = 0.02x₁
0.01x₁² -2x₁+50 = 0
Using the quadratic formula,
x₁ = 29.3 or 170.7
170.7 is invalid because then the statue is behind the car, so
x₁ = 29.3
y₁ = (0.01)29.3² = 8.6
The headlights hit the statue when the car is at (29.3, 8.6).