I have something I wanna ask about. If a plane flies along curve 1/x^2 it shoots at a bunker at position (10,0). The shot will follow the tangent line of the curve. For what value x= a should the pilot shoot to get direct hit on bunker?
I am a little bit unsure what to do here. I don't understand by the question if the tangent of the curve, is the tangent in point (10, 0). I guess not, since the curve 1/x^2 never reaches that point. I guess if the tangent reaches (10, 0) I must draw a tangent that hits curve and also the bunker. Then follow point of tangent down to x-axis, call that point a. Follow the point to the y-axis and then call that point L. I guess I could use the limit and see that i must be within an area epsilon around L, and delta around a, to be able to hit the bunker? They want us to calculate a, as position from where pilot should shoot. Honestly I have no idea how to do it with just a position of the bunker at (10, 0) and curve that pilot follows, f(x) = 1/x^2.
Anyone who can help me get started on this?? Maybe give me some clues and some good tips :D pleease do :)
Assume that the plane shoots at the point $ (x_0,\frac{1}{x_0^2})$.
Now, this bullet must reach the bunker at (10,0) . Since,path is a straight line along the tangent , (10,0) satisfies the equation of tangent at point $ (x_0,\frac{1}{x_0^2})$.
Therefore,,$$\frac{\frac{1}{x_0^2}-0}{x_0-10} = f'(x)=\frac{-2}{x_0^3}$$ Solving this ,gives $x_0 = \frac{20}{3}.$