Chapter 8.3, Problem 6GP

Solution

To calculate : The focal length (the distance from the vertex to the focus).

  $8.4$ miles west he will be in the range of tower.

Given information :

A cellular phone tower services a $10$ miles radius. A person gets a flat tire $4$ miles east and $9$ miles north of the tower. He fixes the tire and then drives south.

Calculation :

Radius of the cellular phone tower is $10$ miles.

So, the equation is $x^2+y^2={10}^2$ .

There is a flat tire $4$ miles east and $9$ miles north to the tower. Order the pair as $\left(4,9\right)$ .

Suppose the driver drives west after fixing the tire. When the drives leaves the tower’s range, he will be at a point on the circle $x^2+y^2={10}^2$ whose $x-$ coordinates is negative and whose $y-$ coordinates is $9$ .

Find the point $\left(x,9\right)$ , where $x<0$ on the circle $x^2+y^2={10}^2$ .

  $\begin{array}{l}{x}^2+y^2={10}^2\\ x^2+9^2={10}^2\\ x^2+81=100\\ x^2=19\\ x=\pm 4.4\end{array}$

Because $x<0$ , $x=-4.4$ . You will be in the towers range from $\left(4,9\right)\text{ to }\left(-4.4,9\right)$ .

Find the distance between $x-$ coordinates of point $\left(4,9\right)\text{ to }\left(-4.4,9\right)$ .

  $\begin{array}{c}\left|4-\left(-4,4\right)\right|=\left|4+4.4\right|\\ =\left|8.4\right|\\ =8.4\end{array}$

Therefore, $8.4$ miles west he will be in the range of tower.