Chapter 2, Problem 14P

To determine

To find:The point on the highway will the car’s headlights illuminate the status.

Answer to Problem 14P

The point on the highway will the car’s headlights illuminate the status at $\left(100-50\sqrt{2}\right)\text{m}$ east and $\left(150-100\sqrt{2}\right)\text{m}$ north to the origin.

Explanation of Solution

Given information:

The car travelling along a highway shaped like parabola with its vertex at the origin as shown in figure (1).

  

Figure (1)

Calculation:

The parabola has the vertex at $\left(0,0\right)$ and has $y$ -axis as its axis.it shows that the parabola is in the upper half plane and the equation of the parabola will be $y=a{x}^2$ with $a>0$ .

The point $\left(-100,100\right)$ is on the parabola.

  $\begin{array}{c}y=a{x}^2\\ 100={\left(100\right)}^2\\ a=\frac{1}{100}\end{array}$

So, the equation of parabola is.

  $y=\frac{x^2}{100}$

Assume that the headlight of the car illuminates the statue at the point of coordinate $\left(x_0,y_0\right)$ .

  $y_0=\frac{x_0^2}{100}$

The coordinate of the statue is $\left(100,50\right)$ , the tangent line to the parabola at point $\left(x_0,y_0\right)$ passes through the point $\left(100,50\right)$ .

Calculate the slope of the tangent line at any point is.

  $\begin{array}{c}y\text{'}=\frac{2x}{100}\\ =\frac{x}{50}\end{array}$

Theslope of the tangent line at $\left(x_0,y_0\right)$ is.

  $y\text{'}=\frac{x_0}{50}$

The equation of tangent line is.

  $\left(y-y_0\right)=\frac{x_0}{50}\left(x-x_0\right)$

It passes through the point $\left(100,50\right)$ .

  $\begin{array}{c}\left(50-y_0\right)=\frac{x_0}{50}\left(100-x_0\right)\\ 50-y_0=2x_0-\frac{x_0^2}{50}\end{array}$

As we know that, $y_0=\frac{x_0^2}{100}$ .

  $\begin{array}{c}50-y_0=2x_0-2y_0\\ y_0=2x_0-50\end{array}$

Substitute $\frac{x_0^2}{100}$ for $y_0$ above linear equation.

  $\begin{array}{c}\frac{x_0^2}{100}=2x_0-50\\ x_0^2-200x_0+5000=0\\ x_0=100\pm 50\sqrt{2}\end{array}$

The minus sign is the right one, and the other one corresponds to when the taillights of the car would be illuminating the statue.

The value of $y_0$ will be.

  $\begin{array}{c}{y}_0=2\left(100-50\sqrt{2}\right)-50\\ =200-100\sqrt{2}-50\\ =150-100\sqrt{2}-50\end{array}$

Therefore, the point on the highway will the car’s headlights illuminate the status at $\left(100-50\sqrt{2}\right)\text{m}$ east and $\left(150-100\sqrt{2}\right)\text{m}$ north to the origin.