Chapter 8.1, Problem 55PS

Solution

To calculate : The diameter of the crater.

The diameter of the crater is $550\text{ feet}$ .

Given information :

Five meteor craters are clustered together near Odessa, Texas. Three points on edge of the circular main crater can be represented by $\left(-220,220\right),\left(0,0\right),\text{ and }\left(200,40\right)$ where each unit represents $1$ foot.

Formula used:

The distance $\left(d\right)$ between two points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ is given by,

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

Calculation :

Let $\left(x,y\right)$ be the center of the crater, then the distance $\left(d_1\right)$ between $\left(x,y\right)$ and $\left(0,0\right)$ will be,

  $\begin{array}{c}{d}_1=\sqrt{{\left(0-x\right)}^2+{\left(0-y\right)}^2}\\ =\sqrt{x^2+y^2}\end{array}$

The distance $\left(d_2\right)$ between $\left(x,y\right)$ and $\left(-220,220\right)$ will be,

  $\begin{array}{c}{d}_2=\sqrt{{\left(-220-x\right)}^2+{\left(220-y\right)}^2}\\ =\sqrt{48400+440x+x^2+48400-440y+y^2}\\ =\sqrt{x^2+440x-440y+y^2+96800}\end{array}$

The distance $\left(d_3\right)$ between $\left(x,y\right)$ and $\left(200,40\right)$ will be,

  $\begin{array}{c}{d}_3=\sqrt{{\left(200-x\right)}^2+{\left(40-y\right)}^2}\\ =\sqrt{40000-400x+x^2+1600-80y+y^2}\\ =\sqrt{x^2-400x+41600-80y+y^2}\end{array}$

As, $d_1,d_2,\text{ and }{d}_3$ are the distance from center; so they are the radius of the crater and thus $d_1=d_2=d_3$ .

Equate $d_1\text{ and }{d}_2$ .

  $\begin{array}{c}\sqrt{x^2+y^2}=\sqrt{x^2+440x-440y+y^2+96800}\\ x^2+y^2=x^2+440x-440y+y^2+96800\left[\text{Square both the sides}\right]\\ x^2+y^2-x^2-y^2=440x-440y+96800\\ 440\left(x-y+220\right)=0\\ x-y+220=0\\ x-y=-220\mathrm{...............}\left(1\right)\end{array}$

Equate $d_1\text{ and }{d}_3$ .

  $\begin{array}{c}\sqrt{x^2+y^2}=\sqrt{x^2-400x+41600-80y+y^2}\\ x^2+y^2=x^2-400x+41600-80y+y^2\left[\text{Square both the sides}\right]\\ x^2+y^2-x^2-y^2=-400x+41600-80y\\ -80\left(5x+y-520\right)=0\\ 5x+y-520=0\\ 5x+y=520\mathrm{...............}\left(2\right)\end{array}$

Add (1) and (2)

  $\begin{array}{c}6x=300\\ x=50\end{array}$

Substitute the value of $x$ in (1)

  $\begin{array}{c}x-y=-220\\ 50-y=-220\\ y=270\end{array}$

Diameter is twice the radius. So, substitute the value of $x\text{ and }y$ in $d_1$ to find the diameter $\left(D\right)$ .

  $\begin{array}{c}D=2\left(\sqrt{{\left(50\right)}^2+{\left(270\right)}^2}\right)\\ =2\left(\sqrt{2500+72900}\right)\\ =2\left(\sqrt{75400}\right)\\ =2\left(274.59\right)\\ \approx 550\text{ feet}\end{array}$

Therefore, the diameter of the crater is $550\text{ feet}$ .