you are helping your city plan emergency routes in a new neighborhood. the neighborhood is on a coordinate grid where each square represents 100 meters. emergency vehicles can travel on streets (grid lines) or cut diagonally through parks when responding to emergencies. question 4 the city wants to place a new emergency center at (6, 3). calculate the distance from this center to each of the three existing emergency services. is this a good central location?

Question

Accepted Answer

# Explanation:
## Step1: Find coordinates of existing services
Assume the coordinates of the three existing services are $ (2,1) $ (fire - truck), $ (5,7) $ (police), $ (8,2) $ (hospital). The new center is at $ (6,3) $. Use the distance formula $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $.
## Step2: Calculate distance to fire - truck
For points $ (6,3) $ and $ (2,1) $:
$$
\begin{align*}
d_1&=\sqrt{(6 - 2)^2+(3 - 1)^2}\
&=\sqrt{4^2+2^2}\
&=\sqrt{16 + 4}\
&=\sqrt{20}\
&=2\sqrt{5}\approx4.47
\end{align*}
$$
Since each square is $ 100 $ meters, the actual distance $ D_1 = 4.47\times100=447 $ meters.
## Step3: Calculate distance to police
For points $ (6,3) $ and $ (5,7) $:
$$
\begin{align*}
d_2&=\sqrt{(6 - 5)^2+(3 - 7)^2}\
&=\sqrt{1^2+(-4)^2}\
&=\sqrt{1 + 16}\
&=\sqrt{17}\approx4.12
\end{align*}
$$
The actual distance $ D_2=4.12\times100 = 412 $ meters.
## Step4: Calculate distance to hospital
For points $ (6,3) $ and $ (8,2) $:
$$
\begin{align*}
d_3&=\sqrt{(6 - 8)^2+(3 - 2)^2}\
&=\sqrt{(-2)^2+1^2}\
&=\sqrt{4+1}\
&=\sqrt{5}\approx2.24
\end{align*}
$$
The actual distance $ D_3=2.24\times100=224 $ meters.

# Answer:
The distances from the new emergency center $(6,3)$ to the three existing services are approximately $447$ meters (to the fire - truck), $412$ meters (to the police), and $224$ meters (to the hospital). Whether it is a good central location depends on the context. If relatively balanced distances to all three services are desired, and considering the distances are not extremely disparate (for example, if the maximum distance $447$ meters is acceptable compared to the minimum $224$ meters in the context of the city's emergency response requirements), it could be a reasonable central location. But if a more equal - distance distribution is preferred (e.g.,  if the city wants all distances to be within a very narrow range), it may not be the optimal one.

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QUESTION IMAGE

Question

Explanation:

Step1: Find coordinates of existing services

Step2: Calculate distance to fire - truck

Step3: Calculate distance to police

Step4: Calculate distance to hospital

Answer: