Question

time capsule graduating classes have buried time capsules on the campus of east side high school for over 20 years. the points on the diagram show the position of three time capsules. find the distance between each pair of time capsules.

1. a(4, 9), b(2, - 3)
2. a(4, 9), c(9, 0)
3. b(2, - 3), c(9, 0)
4. mp structure which two time capsules are the closest to

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate distance between A(4, 9) and B(2, - 3)

Let $(x_1,y_1)=(4,9)$ and $(x_2,y_2)=(2,-3)$. Then $d_{AB}=\sqrt{(2 - 4)^2+(-3 - 9)^2}=\sqrt{(-2)^2+(-12)^2}=\sqrt{4 + 144}=\sqrt{148}=2\sqrt{37}\approx 12.1655$.

Step3: Calculate distance between A(4, 9) and C(9, 0)

Let $(x_1,y_1)=(4,9)$ and $(x_2,y_2)=(9,0)$. Then $d_{AC}=\sqrt{(9 - 4)^2+(0 - 9)^2}=\sqrt{5^2+(-9)^2}=\sqrt{25 + 81}=\sqrt{106}\approx 10.2956$.

Step4: Calculate distance between B(2, - 3) and C(9, 0)

Let $(x_1,y_1)=(2,-3)$ and $(x_2,y_2)=(9,0)$. Then $d_{BC}=\sqrt{(9 - 2)^2+(0+3)^2}=\sqrt{7^2+3^2}=\sqrt{49 + 9}=\sqrt{58}\approx 7.6158$.

Step5: Compare distances

We have $d_{BC}\approx7.6158$, $d_{AC}\approx10.2956$, $d_{AB}\approx12.1655$. So B and C are closest, A and B are farthest.

Answer:

1. $d_{AB}=2\sqrt{37}\approx 12.1655$
2. $d_{AC}=\sqrt{106}\approx 10.2956$
3. $d_{BC}=\sqrt{58}\approx 7.6158$
4. Closest: B and C; Farthest: A and B