Chapter 1.3, Problem 6CYU

To determine

Two find the closest and farthest time capsules.

Answer to Problem 6CYU

AB is farthest and BC is closest.

Explanation of Solution

Given information:

Point A, B and C.

A graph

Formula used:

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

Calculation:

Observing the graph,

Coordinates of $A=(4,9)$

Coordinates of $B=(2,-3)$

Coordinates of $C=(9,0)$

Using distance formula,

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

For AB,

  $x_2^{}=2,x_1^{}=4$ and $y_2^{}=-3,y_1^{}=9$

Substituting values,

AB= $\sqrt{{(2-4)}^2+{(-3-9)}^2}$

Solving,

AB= $\sqrt{{(-2)}^2+{(-12)}^2}$

Hence, distance between AB is $\sqrt{148}$ .

For BC,

  $x_2^{}=9,x_1^{}=2$ and $y_2^{}=0,y_1^{}=-3$

Substituting values,

BC= $\sqrt{{(9-2)}^2+{(0-(-3))}^2}$

Solving,

BC= $\sqrt{{(7)}^2+{(3)}^2}$

Hence, distance between BC is $\sqrt{58}$ .

For CA,

  $x_2^{}=4,x_1^{}=9$ and $y_2^{}=9,y_1^{}=0$

Substituting values,

CA= $\sqrt{{(4-9)}^2+{(9-0)}^2}$

Solving,

CA= $\sqrt{{(-5)}^2+{(9)}^2}$

Hence, distance between CA is $\sqrt{106}$ .

Looking at the values, it is observed that, AB is farthest and BC is closest.