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October 17th, 2007, 02:28 PM   #1
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calc help

A car is traveling at night along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100 m west and 100 m north of the origin and travels inan easterly direction. There is a statue located 100 m east and 50 m north of the origin. At what point on the highway will the car's headlights illuminate the statue?

what i have so far:
y'(x)=m=(50-y)/(100-x)
delta x=100-x
delta y=50-y
delta y=f(x+deltax)-f(x)
=50-y=100-x
y'=1
y=x-50

I dont no how to finish the problem off so and i feel like i am going in circles, so any help would be great! Thanks!!!!
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October 18th, 2007, 06:57 AM   #2
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If we assume that the car's range of lights is a straight line, then your question is: find the coordinates of the parabola mentioned in your text what, if we put a tangent to the parabola in that coordinates, we will get a line that will have the point of the statue as a solution! So, first, we have to find the function of the car's path. We have:

f(x)=a*x^2+b*x+c
f(0)=0 -> vector is in the origin (1)
f'(0)=0 -> vector is in the origin (2)
f(-100)=100 -> the car's original point of starting

from the first data we get:

a*0^2+b*0+c=0 -> c=0

Now:

f(x)=a*x^2+b*x

from the second data we get:

f'(x)=2a*x+b
f'[0]=b=0

Again:

f(x)=a*x^2

and from the third data we get:

f8-100)=a*100^2=100 -> a=1/100

Finally:

f(x)=1/100*x^2

Now, the formula for a tangent to a function through a point (x0,y0) is:

y-y0=f'(x0)(x-x0)

f'(x0)=x0/50

and since one of the solutions of the tangent has to be (100,50), we get:

50-y0=x0/50(100-x0) (*)

finally, since the tangent has to touch the parabola in x0, we can get:

f(x0)=y0=x0^2/100

plugging that in (*) we get:

50-x0^2/100=x0/50(100-x0)
50-x0^2/100=2*x0-x0^2/50
x0^2/100-2x0+50=0
x0^2-200x0+5000=0

solving for x0 we get:

x0=50*(2-sqrt(2))
and
x0=50*(2+sqrt(2))

the reason we get two point is justified, but our solution is only one, and that's:

x0=50*(2-sqrt(2))

The reason for this is the fact that the car is traveling from the west to the east, and the light are in front of the car. If it would have been the other way, our solution would have been the other one!

Now, finally, the point we are looking for is:

(x0,f(x0))=(50*(2-sqrt(2)),25(2-sqrt(2))^2)
milin is offline  
October 18th, 2007, 09:40 AM   #3
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Does that solution really satisfy the requirement that the car initially travels in an easterly direction?
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October 18th, 2007, 09:53 AM   #4
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Well, I think so.. Would you disagree?
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October 20th, 2007, 03:40 AM   #5
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I would. It starts 100 m west and 100 m north of the origin, yet it reaches the origin. On a straight line route, it would be going south-east. For the parabola considered, the initial direction was much nearer south than east.
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October 21st, 2007, 09:00 AM   #6
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This is how I pictured it! The red arrow represents the original car direction, the green one the solution I find to be true, and the blue one the one I find to be wrong! (The parabola, of course represents the car's path)

[/url]
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