To solve this problem, we can use the law of cosines. The law of cosines is particularly helpful when dealing with non-right triangles, and it states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] and the angle [tex]\(\gamma\)[/tex] opposite side [tex]\(c\)[/tex]:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\gamma) \][/tex]
Here's a step-by-step solution to the problem:
1. Identify the sides of the triangle:
- You walk 55 meters to the north. Let's denote this side as [tex]\( a \)[/tex].
- Then, you turn 60° to your right and walk another 45 meters. Let's denote this side as [tex]\( b \)[/tex].
2. Determine the angle between the two sides:
- The angle between the direction you first walked (north) and the direction after you turned to your right (60° to the east of north) is [tex]\( 60° \)[/tex]. Let's denote this angle as [tex]\( \gamma \)[/tex].
3. Apply the law of cosines:
- Using the law of cosines, we need to find [tex]\( c \)[/tex] (the distance from your starting point to your ending point):
[tex]\[ c = \sqrt{a^2 + b^2 - 2ab \cos(\gamma)} \][/tex]
4. Substitute the values:
- [tex]\( a = 55 \)[/tex] meters
- [tex]\( b = 45 \)[/tex] meters
- [tex]\( \gamma = 60° \)[/tex]
[tex]\[ c = \sqrt{55^2 + 45^2 - 2 \cdot 55 \cdot 45 \cdot \cos(60°)} \][/tex]
5. Calculate the cosine of 60°:
- [tex]\(\cos(60°) = 0.5\)[/tex]
6. Substitute [tex]\(\cos(60°)\)[/tex] into the equation:
[tex]\[ c = \sqrt{55^2 + 45^2 - 2 \cdot 55 \cdot 45 \cdot 0.5} \][/tex]
7. Compute the values inside the square root:
[tex]\[ 55^2 = 3025 \][/tex]
[tex]\[ 45^2 = 2025 \][/tex]
[tex]\[ 2 \cdot 55 \cdot 45 \cdot 0.5 = 2475 \][/tex]
8. Combine these values:
[tex]\[ c = \sqrt{3025 + 2025 - 2475} \][/tex]
[tex]\[ c = \sqrt{ 5050 - 2475 } \][/tex]
[tex]\[ c = \sqrt{2575} \][/tex]
9. Find the square root:
[tex]\[ c \approx 50.74445782546109 \][/tex]
Therefore, the distance from where you originally started is approximately:
[tex]\[ 50.74445782546109 \text{ meters} \][/tex]
So, the closest answer is:
[tex]\[ \boxed{50 \text{ meters}} \][/tex]
