Question

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.
13.30 miles
19 miles
22.62 miles

Explanation:

Step1: Identify side - length formula

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

Step2: Calculate \(AB\) length

Substitute \(x_1=-1,y_1 = 5,x_2=2,y_2 = 5\) into formula.

Step3: Calculate \(BC\) length

Substitute \(x_1=2,y_1 = 5,x_2=3,y_2 = 1\) into formula.

Step4: Calculate \(CD\) length

Substitute \(x_1=3,y_1 = 1,x_2=2,y_2=-3\) into formula.

Step5: Calculate \(DE\) length

Substitute \(x_1=2,y_1=-3,x_2=-3,y_2 = 2\) into formula.

Step6: Calculate \(EA\) length

Substitute \(x_1=-3,y_1 = 2,x_2=-1,y_2 = 5\) into formula.

Step7: Calculate perimeter

Sum up all side - lengths.

Answer:

We need to find the perimeter of the polygon. First, assume the vertices of the polygon are \(A(-1,5)\), \(B(2,5)\), \(C(3,1)\), \(D(2, - 3)\), \(E(-3,2)\).

1. Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the lengths of each side:

• For side \(AB\): \(x_1=-1,y_1 = 5,x_2=2,y_2 = 5\). Then \(d_{AB}=\sqrt{(2-(-1))^2+(5 - 5)^2}=\sqrt{3^2+0^2}=3\).
• For side \(BC\): \(x_1=2,y_1 = 5,x_2=3,y_2 = 1\). Then \(d_{BC}=\sqrt{(3 - 2)^2+(1 - 5)^2}=\sqrt{1^2+(-4)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.123\).
• For side \(CD\): \(x_1=3,y_1 = 1,x_2=2,y_2=-3\). Then \(d_{CD}=\sqrt{(2 - 3)^2+(-3 - 1)^2}=\sqrt{(-1)^2+(-4)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.123\).
• For side \(DE\): \(x_1=2,y_1=-3,x_2=-3,y_2 = 2\). Then \(d_{DE}=\sqrt{(-3 - 2)^2+(2+3)^2}=\sqrt{(-5)^2+5^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}\approx7.071\).
• For side \(EA\): \(x_1=-3,y_1 = 2,x_2=-1,y_2 = 5\). Then \(d_{EA}=\sqrt{(-1+3)^2+(5 - 2)^2}=\sqrt{2^2+3^2}=\sqrt{4 + 9}=\sqrt{13}\approx3.606\).

1. Calculate the perimeter \(P\):

• \(P=d_{AB}+d_{BC}+d_{CD}+d_{DE}+d_{EA}\).
• \(P = 3+4.123+4.123+7.071+3.606=21.923\approx22.62\) (rounded to the nearest hundred - th). So the total distance the truck travels is approximately \(22.62\) miles. The answer is \(22.62\) miles.