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 December 2nd, 2010, 12:23 PM #1 Newbie   Joined: Dec 2010 Posts: 2 Thanks: 0 Help with calculus question (car's headlights statue) I've been staring at this problem for 2 hours and I still have nothing to show for it. I'm really terrible at math and I've been struggling with calculus all year so far. Here's the question: "You are driving a car at night along a highway shaped like a parabola with its vertex at the origin. the car starts at a point 100 meters west and 100 meters north of the origin and travels in an easterly direction. there is a statue located 100 meters east and 50 meters noth of the origin. At what point on the highway will the car's headlights illuminate the statue? Hint: the equation of the parabola is y=1/100 x^2." He told us to use this formula: x = -b ± sqrt( b2-4ac ) ----------------------------------------------------- _______________ ------------------------------------------------------------ 2a I know all these hints should help me, but like I said before I'm completely hopeless when it comes to math. Help would be greatly appreiciated.
 December 2nd, 2010, 03:19 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Help with calculus question (car's headlights statue) Essentially what you need to do is find the point $(x,y)$ on the parabola where its tangent line passes through the coordinates of the statue $(x_S,y_S)$ and where $(x,y)<(x_S,y_S)$ since the headlights point in the direction of travel and the car is coming from the west. To begin, we can set the slope of the line segment with terminals at the car and the statue equal to the slope of the parabola $\frac{dy}{dx}=\frac{x}{50}$: $\frac{y_S-y}{x_S-x}=\frac{x}{50}$ Cross multiply: $50y_S-50y=x_Sx-x^2$ Use $y=\frac{x^2}{100}$ $50y_S-\frac{x^2}{2}=x_Sx-x^2$ Put quadratic into standard form: $\frac{x^2}{2}-x_Sx+50y_S=0$ Multiply through by 2: $x^2-2x_Sx+100y_S=0$ Apply quadratic formula to find x (take smaller root): $x=\frac{2x_S-\sqrt{4x_S^2-400y_S}}{2}=x_S-\sqrt{x_S^2-100y_S}$ Plug in given coordinates of statue: $x=100-\sqrt{100^2-100\cdot 50}=100-50\sqrt{2}=50\left(2-\sqrt{2}\right)$ $y=\frac{x^2}{100}=\frac{\left(50\left(2-\sqrt{2}\right)\right)^2}{100}=50\left(3-2\sqrt{2}\right)$ Thus, the car will be approximately 29.29 m. east of the vertex and 8.58 m. north of the vertex when its headlights illuminate the statue.
 December 2nd, 2010, 04:13 PM #3 Newbie   Joined: Dec 2010 Posts: 2 Thanks: 0 Re: Help with calculus question (car's headlights statue) I can't thank you enough for your help. Thank you, thank you, thank you, SO much.

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# a car takes the shape of y=x^2 and its headlights illuminate a deer at point 1, 1/2 . find the location of the car

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