Chapter 8.3, Problem 64PS

Solution

To find: the house located outside the free delivery area.

D.

Given information:

The appliance store provides free delivery up to 100 miles radius. The locations of the houses are:

1. $\left(95,30\right)$
2. $\left(90,35\right)$
3. $\left(80,55\right)$
4. $\left(75,70\right)$

Formula Used:

The Distance Formula

The distance d between $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is:

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Standard equation of a circle with Center at the Origin

The standard form of the equation of a circle with center at $\left(0,0\right)$ and radius r is as follows:

  $x^2+y^2=r^2$

Explanation:

Let the position of the appliance store is at Origin $\left(0,0\right)$ .

Now, it is given that the store provides free delivery up to 100 miles radius. So, the region of free delively is the circular area around the store with 100 miles as radius.

  $\begin{array}{c}{x}^2+y^2={100}^2\\ x^2+y^2=10000\end{array}$

The distance between the tower and the house A $\left(95,30\right)$ is:

  $\begin{array}{c}{d}_1=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(95-0\right)}^2+{\left(30-0\right)}^2}\\ =\sqrt{9025+900}\\ =\sqrt{9925}\end{array}$

Now, as $9925<10000$ , so the house A is located within the free delivery area.

The distance between the tower and the house B $\left(90,35\right)$ is:

  $\begin{array}{c}{d}_2=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(90-0\right)}^2+{\left(35-0\right)}^2}\\ =\sqrt{8100+1225}\\ =\sqrt{9325}\end{array}$

Now, as $9325<10000$ , so the house B is located within the free delivery area.

The distance between the tower and the house C $\left(80,55\right)$ is:

  $\begin{array}{c}{d}_3=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(80-0\right)}^2+{\left(55-0\right)}^2}\\ =\sqrt{6400+3025}\\ =\sqrt{9425}\end{array}$

Now, as $9425<10000$ , so the house C is located within the free delivery area.

The distance between the tower and the house D $\left(75,70\right)$ is:

  $\begin{array}{c}{d}_4=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(75-0\right)}^2+{\left(70-0\right)}^2}\\ =\sqrt{5625+4900}\\ =\sqrt{10525}\end{array}$

Now, as $10525>10000$ , so the house D is located outside the free delivery area.