Question

equivalent experience_ma.912.gr.1.3_unit 6
two groups of cyclists leave from the same starting point.
group a rides 12 miles east, turns 45° north of east, and then rides 8 miles.
group b rides 12 miles west, turns 30° north of west, and then rides 8 miles.
which of the following is a true statement about who is closer to the starting point?
○ group a is closer.
○ group b is closer.
○ the groups of the same distance
○ there is not enough information to tell.

Explanation:

Step1: Analyze the triangle for Group A

For Group A, we can model their path as a triangle with sides \( a = 12 \), \( b = 8 \), and the included angle \( \theta_A=180^{\circ}- 45^{\circ}=135^{\circ} \) (since they turned \( 45^{\circ} \) north of east, the angle between the two paths is \( 180 - 45=135^{\circ} \)). We use the Law of Cosines \( c^2=a^2 + b^2-2ab\cos\theta \) to find the distance from the starting point (\( c_A \)).

\( c_A^2=12^2 + 8^2-2\times12\times8\times\cos(135^{\circ}) \)

\( \cos(135^{\circ})=-\frac{\sqrt{2}}{2}\approx - 0.7071 \)

\( c_A^2 = 144 + 64-2\times12\times8\times(- 0.7071) \)

\( c_A^2=208 + 135.2832=343.2832 \)

\( c_A=\sqrt{343.2832}\approx18.53 \)

Step2: Analyze the triangle for Group B

For Group B, their path is a triangle with sides \( a = 12 \), \( b = 8 \), and the included angle \( \theta_B = 180^{\circ}-30^{\circ}=150^{\circ} \) (since they turned \( 30^{\circ} \) north of west, the angle between the two paths is \( 180 - 30 = 150^{\circ} \)). Using the Law of Cosines to find \( c_B \) (distance from starting point):

\( c_B^2=12^2+8^2 - 2\times12\times8\times\cos(150^{\circ}) \)

\( \cos(150^{\circ})=-\frac{\sqrt{3}}{2}\approx - 0.8660 \)

\( c_B^2=144 + 64-2\times12\times8\times(-0.8660) \)

\( c_B^2=208+165.888 = 373.888 \)

\( c_B=\sqrt{373.888}\approx19.33 \)

Step3: Compare \( c_A \) and \( c_B \)

Since \( c_A\approx18.53 \) and \( c_B\approx19.33 \), and \( 18.53<19.33 \), Group A is closer to the starting point.

Answer:

Group A is closer.