Question

date: sept22,29 hour: 6 1/2 cameron, arthur, and jamie are playing soccer. their locations are recorded by a motion tracking system. the distance between grids is 5 meters. (hint: find their coordinates first or make the right triangles!) 20. how far apart are arthur and jamie? round to the nearest tenth of a meter. d = 21. who is closer to cameron? explain. 22. the soccer ball is located at the point (35, 60). who is closest to the soccer ball?

Explanation:

Step1: Identify coordinates

Arthur's coordinates are $(20,35)$, Jamie's coordinates are $(45,20)$.

Step2: Use distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 20,y_1 = 35,x_2 = 45,y_2 = 20$.
So $d=\sqrt{(45 - 20)^2+(20 - 35)^2}=\sqrt{25^2+(- 15)^2}=\sqrt{625 + 225}=\sqrt{850}\approx29.2$ meters.

Step3: Distance between Cameron and Arthur

Cameron's coordinates are $(65,40)$. Using the distance formula for Arthur $(20,35)$ and Cameron $(65,40)$:
$d_{AC}=\sqrt{(65 - 20)^2+(40 - 35)^2}=\sqrt{45^2+5^2}=\sqrt{2025 + 25}=\sqrt{2050}\approx45.3$ meters.

Step4: Distance between Cameron and Jamie

Using the distance formula for Jamie $(45,20)$ and Cameron $(65,40)$:
$d_{JC}=\sqrt{(65 - 45)^2+(40 - 20)^2}=\sqrt{20^2+20^2}=\sqrt{400+400}=\sqrt{800}\approx28.3$ meters. Since $28.3<45.3$, Jamie is closer to Cameron.

Step5: Distance between the ball and Arthur

Ball's coordinates are $(35,60)$. Using the distance formula for Arthur $(20,35)$ and the ball $(35,60)$:
$d_{AB}=\sqrt{(35 - 20)^2+(60 - 35)^2}=\sqrt{15^2+25^2}=\sqrt{225 + 625}=\sqrt{850}\approx29.2$ meters.

Step6: Distance between the ball and Jamie

Using the distance formula for Jamie $(45,20)$ and the ball $(35,60)$:
$d_{JB}=\sqrt{(35 - 45)^2+(60 - 20)^2}=\sqrt{(-10)^2+40^2}=\sqrt{100 + 1600}=\sqrt{1700}\approx41.2$ meters.

Step7: Distance between the ball and Cameron

Using the distance formula for Cameron $(65,40)$ and the ball $(35,60)$:
$d_{CB}=\sqrt{(35 - 65)^2+(60 - 40)^2}=\sqrt{(-30)^2+20^2}=\sqrt{900+400}=\sqrt{1300}\approx36.1$ meters. Since $29.2<36.1<41.2$, Arthur is closest to the soccer - ball.

Answer:

1. $29.2$ meters
2. Jamie is closer to Cameron because the distance between Jamie and Cameron ($d_{JC}\approx28.3$ meters) is less than the distance between Arthur and Cameron ($d_{AC}\approx45.3$ meters).
3. Arthur is closest to the soccer - ball.