I need help with this application.
A pier 1250 meters long extends at an angle from the shoreline. A surveyor walks to a point 1500 meters down the shoreline from the pier and measures the angle formed by the ends of the pier. It is found to be 53 degrees.
What acute angle ( correct to the nearest 0.1 degree) does the pier form with the shoreline? Is there more than one possibility? If so, how can we know which is the correct one?
Well, well, well, it seems like you're in a bit of a geometric pickle, my friend. Don't worry, Clown Bot is here to help with a dash of humor!
So, let's get down to business. We have a pier, we have a surveyor, and we have some angles going on. The pier extends at an angle from the shoreline, and the surveyor measures an angle formed by the ends of the pier.
Now, to find out the acute angle the pier forms with the shoreline, we need to tap into our trigonometry knowledge. Remember, trigonometry is just like a secret code that reveals hidden angles.
Let's break it down. We're given the length of the pier (1250 meters) and the distance the surveyor walks down the shoreline (1500 meters). The angle the surveyor measures is 53 degrees.
To find the acute angle the pier forms with the shoreline, we can use the tangent function, which is like the comedic relief of trigonometry. Tangent is equal to the opposite side divided by the adjacent side.
In our clownish scenario, the opposite side is the length of the pier, and the adjacent side is the distance the surveyor walks down the shoreline. So, we have:
tan(Angle) = Pier length / Distance walked
tan(Angle) = 1250 / 1500
Now, to find the acute angle, we need to take the inverse tangent (atan) of the right-hand side of our equation. And voila, we have our answer!
The acute angle the pier forms with the shoreline is approximately 40.3 degrees.
Now, to address the second part of your question, if you're lucky, there might be only one possibility for the acute angle. But sometimes life throws us a curveball, and there might be more than one possibility. In this case, we have to rely on context or additional information to determine the correct angle.
So, my friend, armed with your newfound trigonometry knowledge and a touch of humor, go forth and conquer that application. Remember, when in doubt, laugh it out!
To find the acute angle that the pier forms with the shoreline, we can use the concept of complementary angles. Complementary angles add up to 90 degrees.
Let's denote the acute angle that the pier forms with the shoreline as 'x' degrees.
Since the surveyor measures the angle formed by the ends of the pier as 53 degrees, we can set up an equation:
x + 53 = 90
By subtracting 53 from both sides of the equation, we have:
x = 90 - 53
x = 37 degrees
So, the acute angle the pier forms with the shoreline is approximately 37 degrees.
Now, we need to determine if there is more than one possibility.
If the surveyor had walked beyond the end of the pier, the angle formed by the ends of the pier would have been obtuse. However, since the angle formed is acute (less than 90 degrees), there is only one possibility for the acute angle that the pier forms with the shoreline.
Therefore, the correct acute angle is 37 degrees.
To find the acute angle that the pier forms with the shoreline, we can use the trigonometric relationships between the sides and angles of a right triangle. In this case, we have a right triangle formed by the pier, the shoreline, and the line connecting the surveyor to the end of the pier.
Let's label the components of the triangle:
- The length of the pier is the hypotenuse (H) of the right triangle.
- The distance from the surveyor to the pier is the adjacent side (A) of the right triangle.
- The distance from the surveyor to the end of the pier is the opposite side (O) of the right triangle.
- The angle formed between the shoreline and the line connecting the surveyor to the end of the pier is 53 degrees.
Now, we can use the trigonometric function tangent (tan) to find the acute angle formed between the pier and the shoreline:
tan(angle) = O / A
In this case, we know O (the distance from the surveyor to the end of the pier) is 1500 meters, and A (the distance from the surveyor to the pier) is unknown. We can rearrange the equation to solve for angle:
angle = arctan(O / A)
To find the value of the unknown A, we can use the Pythagorean theorem:
A^2 + O^2 = H^2
(A^2 + 1500^2 = 1250^2)
Solving this equation will give us the value of A (distance from the surveyor to the pier).
Once we have the value of A, we can substitute it into the equation to find the acute angle formed between the pier and the shoreline:
angle = arctan(1500 / A)
By evaluating this expression, we can determine the acute angle the pier forms with the shoreline.
Is there more than one possibility?
In general, there can be multiple values for the acute angle that satisfy the given conditions. However, in this specific scenario, because we have fixed values for the length of the pier and the distance from the surveyor to the end of the pier, there should only be one possible value for the acute angle formed between the pier and the shoreline.
To verify the correctness of the result, you can double-check the measurements and calculations or repeat the process with different measurements to compare the results.
If the angle you want is a, and the 3rd angle is b, then by the law of sines,
1250/sin53 = 1500/sinb
But, a+b+53=180, so b=127-a
sin(127-a) = 1500sin53/1250 = 0.9583
so, 127-a = 73.4
a = 53.6
I'll let you decide whether there is a second possibility.