Chapter 9, Problem 152RE

To determine

Calculate the distance between the asteroid and earth when $\theta =-\frac{\pi }{3}$ .Also, write the polar equation .

Answer to Problem 152RE

The equation is $r=\frac{12000000}{1-\sin \theta }$ and the distance is $5393876.91miles$ .

Explanation of Solution

Given:

It is given in the question that the asteroid takes a parabolic path with earth as its focus. It is about $6000000$ miles from earth at its closest approach.

Concept Used:

In this, use the concept of parabolic and their properties.

Calculation: Here, the distance between the vertex and the pole is $\frac{p}{2}$ .

  $\begin{array}{l}\frac{p}{2}=6000000\\ p=12000000\end{array}$

Now, it is known that the eccentricity of a parabola is $1$ .

Now, the directrix is horizontal and is below the pole since the vertex is below the pole. Use e and p to write the equation of the parabola.

  $\begin{array}{l}r=\frac{ep}{1-e\sin \theta }\\ r=\frac{12000000}{1-\sin \theta }\end{array}$

Put in $\theta =-\frac{\pi }{3}$ to find r.

  $\begin{array}{l}r=\frac{12000000}{1-\sin \theta }\\ r=\frac{12000000}{1-\sin \frac{-\pi }{3}}\\ r=\frac{12000000}{1-(-\frac{\sqrt{3}}{2})}\\ r\approx 5393876.91\end{array}$

Conclusion:

The equation is $r=\frac{12000000}{1-\sin \theta }$ and the distance is $5393876.91miles$ .