Chapter 8.4, Problem 52PS

Solution

(a)

To explain : The reason that the airplane can fly in the region bounded by the ellipse.

The sum of the distances of all the points from two fixed points called foci is constant.

Given information :

A small airplane with enough fuel to fly $600$ miles safely will take off from airport A and land at airport B, $450$ miles away.

  

Explanation :

By definition, an ellipse is the set of all the points in the plane, the sum of whose distances from two fixed points called foci is constant.

Therefore, the airplane can fly in the region bounded by the ellipse only.

(b)

To Find : The coordinates of each airport.

The coordinates of airport A is $\left(225,0\right)$ and airport B is $\left(-225,0\right)$ .

Given information :

A small airplane with enough fuel to fly $600$ miles safely will take off from airport A and land at airport B, $450$ miles away.

  

Explanation :

The distance between airport A and airport B is $450$ miles.

So,

  $\begin{array}{c}2a=450\\ a=\frac{450}{2}\\ =225\end{array}$

Therefore, the coordinates of airport A is $\left(225,0\right)$ and airport B is $\left(-225,0\right)$ .

(c)

To explain : The coordinates of the vertex on the basis of the distance the plane flies.

The coordinates of the vertex on the basis of the distance the plane flies are $\left(300,0\right)$ and $\left(-300,0\right)$ .

Given information :

A small airplane with enough fuel to fly $600$ miles safely will take off from airport A and land at airport B, $450$ miles away. Suppose the plane flies from airport A straight past airport B to a vertex of the ellipse and then straight back to airport B.

  

Explanation :

As the plane flies from airport A straight past airport B to a vertex of the ellipse and then straight back to airport B.

The plane can fly maximum $600$ miles, so the vertices of the ellipse formed will be $\left(300,0\right)$ and $\left(-300,0\right)$ .

Therefore, the coordinates of the vertex on the basis of the distance the plane flies are $\left(300,0\right)$ and $\left(-300,0\right)$ .

(d)

To Find : The equation of the ellipse.

The required equation is $\frac{x^2}{{\left(300\right)}^2}+\frac{y^2}{{\left(198.43\right)}^2}=1$ .

Given information :

A small airplane with enough fuel to fly $600$ miles safely will take off from airport A and land at airport B, $450$ miles away.

  

Explanation :

The maximum distance is $600$ .

So,

  $\begin{array}{c}2a=600\\ a=300\\ c=\frac{450}{2}\\ =225\\ b^2=a^2-c^2\\ ={\left(300\right)}^2-{\left(225\right)}^2\\ ={\left(198.43\right)}^2\end{array}$

Therefore, the required equation is $\frac{x^2}{{\left(300\right)}^2}+\frac{y^2}{{\left(198.43\right)}^2}=1$ .