Select the correct answer.

A town park has two paths leading to the playground. One path is 2 miles from the road and the other path makes a right angle with the road. The two paths intersect at an angle of 50∘. What is the approximate distance from the playground to the road along the other path?

 **A.** 

2.6 mi

 **B.** 

1.3 mi

 **C.** 

3.1 mi

 **D.** 

1.5 mi


We are given a scenario with two paths to the playground, and we need to find the distance along the second path. This problem involves using the **law of cosines**, as the paths form a triangle with the road, and we are given an angle and the length of one side.

### Given:
- One path is 2 miles long.
- The two paths intersect at an angle of 50°.
- The other path makes a right angle with the road, so the angle between the second path and the road is \( 90^\circ - 50^\circ = 40^\circ \).
  
We need to find the approximate length of the second path.

### Law of Cosines Formula:
For any triangle, the law of cosines is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

In this case:
- \( c \) is the distance along the other path (which we need to find).
- \( a = 2 \) miles (the known path length).
- \( b = \) the unknown distance.
- \( C = 50^\circ \) (the angle between the paths).

However, it seems some extra information is missing for calculating with precision, such as the length of one of the other sides. Could you provide more details, or clarify any additional measurements?



A town park has two paths leading to the playground. One path is 2 miles from the road and the other path makes a right angle with the road. The two paths intersect at an angle of 50°. What is the approximate distance from the playground to the road along the other path?

A. 2.6 mi

B. 1.3 mi

C. 3.1 mi

D. 1.5 mi

Solve a trigonometric problem involving two paths to a playground, one 2 miles long, intersecting at a 50° angle. Learn how to use the Law of Cosines to find the unknown distance. Suitable for students in Grades 10-12.

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