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In a calculus book, a question reads:

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 in an 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?

I want to know how you would solve this problem, because the method I am using is very round-about. There must be a much more logical way of solving this problem, and I hope you will share it with me. Thank you.

This is my solution..

The function appears to be of the form $$y = f(x) = ax^2$$

$$100 = a(100)^2$$ $$a = 1/100$$ $$y = \frac{x^2}{100}$$

Then I decided (by solving a few tangent equations and generalizing) that the tangent line expression at a point $(p, f(x))$ is given by the function:

$$T(p) = T_m(p)*x + T_b(p)$$

Where $T_m(p)$ represents the slope for a point, $p$ on $f(x)$ and $T_b(p)$ represents the y-intercept value.

$$T_m(p) = \frac{df(p)}{dx} =\frac{p}{50}$$

$$T_b(p) = f(p) - \frac{df(p)}{dx} * p = -\frac{p^2}{100}$$

So given that the statue rests on $(100, 50)$ we know the equation of this tangent line outputs $50$ when $100$ is input.

$$T(p) = \frac{p}{50}*x - \frac{p^2}{100}$$

$$50 = \frac{p}{50}*100 - \frac{p^2}{100}$$

the $x$ position of the car is at a point $p$ such that $p$ satisfies the equation.

I end up getting the positive result $x = 100 - 50\sqrt{2}$

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Note that the problem expects you to assume that the parabola has axis facing straight up. There are piles of parabolas with vertex the origin that face in all sorts of directions. –  André Nicolas Aug 9 '13 at 3:18

1 Answer 1

up vote 3 down vote accepted

Your solution is great. Here's my approach.

We know that the parabola has equation $f(x)=\dfrac{x^2}{100}$. Now suppose that the car illuminates the statue when $x=p$. Then the car's location must be at $\left(p,\frac{p^2}{100}\right)$, and the line connecting this point to $(100,50)$ must have a slope of $f'(p)=\dfrac{p}{50}$. Hence, using the slope formula, we obtain: $$ \begin{align*} \dfrac{\frac{p^2}{100}-50}{p-100} &= \dfrac{p}{50} \\ \dfrac{p^2-5000}{100(p-100)} &= \dfrac{p}{50} \\ \dfrac{p^2-5000}{2(p-100)} &= p \\ p^2-5000 &= 2p(p-100) \\ p^2-5000 &= 2p^2-200p \\ 0 &= p^2-200p+5000 \\ p &= 100 \pm 50\sqrt2 \\ \end{align*} $$

Hence, since we know that the car travels from left to right, we reject the larger solution and conclude that the car illuminates the statue at $x=100-50\sqrt2$.

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I really like the way you approach it, always nice to see things from a more focused perspective. –  Leonardo Aug 9 '13 at 5:24

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