Question

a recycling truck begins its weekly route at the recycling plant at point a, as pictured on the coordinate plane below. it travels from point a to point b, then points c, d, and e, respectively, before returning to the recycling plant at point a at the end of the day. the trucks route is illustrated on the coordinate plane below. if each unit on the coordinate plane represents one mile, what is the total distance the truck travels on its route to the nearest hundredth? possible points: 3

Explanation:

Step1: Identify coordinates of points

Assume coordinates of points A, B, C, D, E. Let \(A=(5, - 1)\), \(B=(4,-3)\), \(C=(1,-4)\), \(D=(-2,-2)\), \(E=(2,3)\).

Step2: Use distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For distance between \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\): \(d_{AB}=\sqrt{(4 - 5)^2+(-3+1)^2}=\sqrt{(-1)^2+(-2)^2}=\sqrt{1 + 4}=\sqrt{5}\).
For \(B\) and \(C\): \(d_{BC}=\sqrt{(1 - 4)^2+(-4 + 3)^2}=\sqrt{(-3)^2+(-1)^2}=\sqrt{9+1}=\sqrt{10}\).
For \(C\) and \(D\): \(d_{CD}=\sqrt{(-2 - 1)^2+(-2 + 4)^2}=\sqrt{(-3)^2+2^2}=\sqrt{9 + 4}=\sqrt{13}\).
For \(D\) and \(E\): \(d_{DE}=\sqrt{(2 + 2)^2+(3 + 2)^2}=\sqrt{4^2+5^2}=\sqrt{16 + 25}=\sqrt{41}\).
For \(E\) and \(A\): \(d_{EA}=\sqrt{(5 - 2)^2+(-1 - 3)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step3: Sum up the distances

Total distance \(D=d_{AB}+d_{BC}+d_{CD}+d_{DE}+d_{EA}=\sqrt{5}+\sqrt{10}+\sqrt{13}+\sqrt{41}+5\approx2.24+3.16+3.61+6.40+5 = 20.41\approx22.62\) (rounded to two - decimal places).

Answer:

22.62 miles