Question: School 2.5 miles 120° 4.5 miles Home [Not drawn to scale] On Monday, Amie rides her bike from…

School

2.5 miles 120° 4.5 miles Home

[Not drawn to scale]

On Monday, Amie rides her bike from home to school. After school, she bikes to work. After work, she bikes home. Based on the information in the diagram, about how many miles does Amie bike on Monday?

12.7 mi. 10.9 mi. 13.1 mi. 11.3 mi.

Solution

To solve the problem, we need to find out the total distance Amie bikes on Monday. The path is from home to school, then school to work, and finally from work back home. 1. Identify the distances: - Home to School: 2.5 miles - School to Work: Need to calculate using the Law of Cosines - Work to Home: 4.5 miles 2. Use the Law of Cosines: For a triangle with sides \( a = 2.5 \) miles, \( b = 4.5 \) miles, and an included angle of \( 120^\circ \), find the distance from School to Work (side \( c \)). The Law of Cosines is given by: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Substitute the known values: \[ c^2 = 2.5^2 + 4.5^2 - 2 \cdot 2.5 \cdot 4.5 \cdot \cos(120^\circ) \] 3. Calculate the cosine: \(\cos(120^\circ) = -0.5\) 4. Plug in the values: \[ c^2 = 6.25 + 20.25 + 2 \cdot 2.5 \cdot 4.5 \cdot 0.5 \] 5. Simplify the equation: \[ c^2 = 26.5 + 11.25 \] \[ c^2 = 37.75 \] 6. Find \( c \): \[ c = \sqrt{37.75} \] \[ c \approx 6.14 \text{ miles} \] 7. Calculate the total distance: Total distance \( = 2.5 + 6.14 + 4.5 \) \[ = 13.14 \text{ miles} \] Therefore, Amie bikes approximately 13.1 miles on Monday.