Bird Flight Problem: Calculating Distance Traveled

Hey guys, let's dive into a cool math problem involving birds and distances! This is a classic trigonometry problem with a real-world twist. We're going to break it down step-by-step so you can see exactly how to solve it. Our scenario involves two birds flying from different points to a common destination, and we need to figure out how far the first bird flew. Sounds fun, right? Let's get started!

Problem Setup

Okay, so here's the situation: We have two birds. Bird 1 starts at point A and flies along a path, let's call it [AC], making a 23° angle with the ground. Bird 2 starts at point B and flies along path [BC], making a 30° angle with the ground. Both birds end up at the same spot, point C. We also know the distance between the starting points, |AB|, is 120 meters. The big question we need to answer is: How far did Bird 1 fly? In other words, what is the length of the path [AC]?

To solve this, we're going to use some trigonometry. Specifically, we'll use the Law of Sines. But before we jump into the math, let's visualize the problem. Imagine a triangle formed by the paths of the birds and the distance between their starting points. This triangle, ABC, is the key to our solution. We know one side (|AB|) and two angles (23° and 30°). To use the Law of Sines effectively, we'll need to find the third angle in the triangle. Remember, the angles in a triangle always add up to 180°.

Finding the Missing Angle

The first crucial step in solving this problem is to determine all the angles within the triangle formed by the birds' flight paths and the ground distance. We already know two angles: the angle at point A (23°) and the angle at point B (30°). To find the third angle, the angle at point C, we use the fundamental property of triangles: the sum of the angles in any triangle is always 180 degrees. This is a core concept in geometry, and it's essential for solving many problems involving triangles. So, let's do the math.

We have Angle A (23°) and Angle B (30°). Let's call Angle C, well, Angle C. The equation looks like this:

Angle A + Angle B + Angle C = 180°

Plugging in the known values:

23° + 30° + Angle C = 180°

Combining the known angles:

53° + Angle C = 180°

Now, to isolate Angle C, we subtract 53° from both sides of the equation:

Angle C = 180° - 53°

Angle C = 127°

So, the angle at point C, where the birds meet, is 127 degrees. This is a significant piece of information because it allows us to use the Law of Sines, which requires knowing at least one side and its opposite angle, and another angle in the triangle. With Angle C calculated, we now have all three angles of the triangle, and we're ready to move on to the next step: applying the Law of Sines to find the distance Bird 1 traveled.

Applying the Law of Sines

Now that we know all three angles of our triangle (23°, 30°, and 127°), and we know the length of one side (|AB| = 120 meters), we can use the Law of Sines to find the distance Bird 1 traveled, which is the length of side [AC]. The Law of Sines is a powerful tool for solving triangles, and it states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. This gives us a set of proportions that we can use to solve for unknown side lengths. The formula for the Law of Sines is:

a / sin(A) = b / sin(B) = c / sin(C)

Where a, b, and c are the side lengths of the triangle, and A, B, and C are the angles opposite those sides, respectively. In our case:

• Side a is [BC] (the distance Bird 2 flew).
• Side b is [AC] (the distance Bird 1 flew – this is what we want to find).
• Side c is [AB] (the distance between the starting points, which is 120 meters).
• Angle A is 23°.
• Angle B is 30°.
• Angle C is 127°.

We want to find side b ([AC]), so we'll set up a proportion using the known side c ([AB]) and its opposite angle (Angle C), and the unknown side b ([AC]) and its opposite angle (Angle B):

b / sin(B) = c / sin(C)

Plugging in the values we know:

[AC] / sin(30°) = 120 / sin(127°)

Now we can solve for [AC].

Solving for the Unknown Distance

Alright, let's get down to the nitty-gritty of solving for the distance Bird 1 traveled, which we've identified as the length of side [AC]. We've already set up the equation using the Law of Sines:

[AC] / sin(30°) = 120 / sin(127°)

The next step is to isolate [AC]. To do this, we'll multiply both sides of the equation by sin(30°). This will get [AC] by itself on the left side of the equation:

[AC] = (120 * sin(30°)) / sin(127°)

Now, we need to calculate the values of the sine functions. You'll probably want to use a calculator for this, especially one that can handle trigonometric functions. Make sure your calculator is in degree mode, not radians, since our angles are in degrees.

• sin(30°) = 0.5 (This is a common trigonometric value that's good to remember!)
• sin(127°) ≈ 0.7986 (This is an approximate value you'll get from your calculator)

Plugging these values into our equation:

[AC] = (120 * 0.5) / 0.7986

Now, let's do the multiplication and division:

[AC] = 60 / 0.7986

[AC] ≈ 75.13 meters

So, we've found that the distance Bird 1 traveled, the length of side [AC], is approximately 75.13 meters. This is the solution to our problem! We used the Law of Sines, along with some basic trigonometry and algebra, to figure out the distance. It's pretty cool how we can use math to solve real-world problems, isn't it?

Conclusion

So, there you have it! By using the Law of Sines and a bit of trigonometry, we've successfully calculated that Bird 1 flew approximately 75.13 meters to reach point C. This problem highlights how useful trigonometry can be in solving real-world scenarios involving angles and distances. I hope this breakdown was helpful and you guys feel a bit more confident tackling similar problems in the future. Remember, the key is to break the problem down into smaller steps, visualize the situation, and apply the appropriate formulas. Keep practicing, and you'll become a math whiz in no time!