Question

emergency response planning
you are helping your city plan emergency routes in a new neighborhood. the neighborhood on a coordinate grid where each square represents 100 meters. emergency vehicles can tra on streets (grid lines) or cut diagonally through parks when responding to emergencies.
question 3
an emergency call comes from location (7,4). calculate how much distance the fire station saves by traveling diagonally instead of following the street grid.

Explanation:

Step1: Find the coordinates of the fire station

Assume the fire station is at \((2,1)\) (from the graph where the fire - truck icon is).

Step2: Calculate the distance along the street grid

The distance along the street grid (using the Manhattan distance formula \(d = |x_2 - x_1|+|y_2 - y_1|\)).
If \((x_1,y_1)=(2,1)\) and \((x_2,y_2)=(7,4)\), then \(d_{grid}=|7 - 2|+|4 - 1|=5 + 3=8\) units. Since each unit is \(100\) meters, the grid - distance \(D_{grid}=8\times100 = 800\) meters.

Step3: Calculate the diagonal distance

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Substitute \(x_1 = 2,y_1 = 1,x_2 = 7,y_2 = 4\) into the formula: \(d=\sqrt{(7 - 2)^2+(4 - 1)^2}=\sqrt{25 + 9}=\sqrt{34}\approx5.83\) units. The diagonal - distance \(D_{diagonal}=5.83\times100=583\) meters.

Step4: Calculate the distance saved

The distance saved \(S=D_{grid}-D_{diagonal}\).
\(S=(8 - 5.83)\times100=(2.17)\times100 = 217\) meters.

Answer:

The fire station saves \(217\) meters.