A surveyor made the two sections of the railroad bridge, both at 210 meters in length. Suppose that the maximum angle of elevation of each section is 75deg. When the bridge is closed, the water level is normally 13 meters below the bridge.
a. When the bridge is fully opened, find the distance from the water to point A on the upper left corner of the right section?
b. When the bridge is fully opened, find the distance between the separated ends of the sections.
a. Well, that's quite a mouthful! But don't worry, I've got this one. So, imagine a bridge opening and closing like a professional contortionist. When the bridge is fully opened, it forms a right-angled triangle. The bridge sections act as the legs of the triangle, and point A is located on the hypotenuse. Using our trusty trigonometry, we can solve this!
Let's start with the right section of the bridge. We know its length is 210 meters and the maximum angle of elevation is 75 degrees. Now, since trigonometry is all about cocktail parties with sine, cosine, and tangent, we'll use a little bit of tangent here.
Using the tangent function, we can say that the tangent of 75 degrees is equal to the opposite side (distance from the water to point A) divided by the adjacent side (length of the right section). Let's call the distance from the water to point A "x".
Now, we can set up our equation:
tan(75) = x / 210
And solving for x, we get:
x = 210 * tan(75)
So, the distance from the water to point A on the upper left corner of the right section is approximately 671.674 meters.
b. Ah, the separation of the bridge ends, like a romantic breakup. When the bridge is fully opened, it forms a straight line. The distance between the separated ends of the sections can be found using simple subtraction.
So, the total length of the bridge (when fully opened) is the sum of the lengths of both sections. Since each section is 210 meters long, the total length is:
210 + 210 = 420 meters.
Therefore, the distance between the separated ends of the sections is 420 meters.
To solve this problem, we can break it down into smaller parts and use trigonometry concepts. Let's start with part A:
a. Finding the distance from the water to point A on the upper left corner of the right section when the bridge is fully opened.
Step 1: Let's draw a diagram to visualize the situation described.
A B
---------------------------------- ----------------------------------
| | | |
Water level | | | |
---------------------------------- ----------------------------------
C D
Here, section AB represents the left section of the bridge, and section BC represents the right section of the bridge. The water level is below section CD.
Step 2: In triangle ABD, we have an angle of 75 degrees at corner A and a side length of 210 meters.
Using trigonometric ratios, we can find the height AD.
Using the trigonometric ratio for tangent:
tan(angle) = opposite/adjacent
tan(75) = AD/210
Solving for AD:
AD = tan(75) * 210
Step 3: Adding the height of the water level, which is normally 13 meters below the bridge, we get the distance from the water to point A on the upper left corner of the right section when the bridge is fully opened.
Distance = AD + 13
b. Finding the distance between the separated ends of the sections when the bridge is fully opened.
Step 1: Let's draw a diagram to visualize the situation described.
A B
---------------------------------- ----------------------------------
| | | |
Water level | | | |
---------------------------------- ----------------------------------
C D
Here, section AB represents the left section of the bridge, and section BC represents the right section of the bridge. The water level is below section CD.
Step 2: In triangle ABD, we have an angle of 75 degrees at corner A and a side length of 210 meters. Similarly, in triangle BCD, we have an angle of 75 degrees at corner B and a side length of 210 meters.
Using trigonometric ratios, we can find the length of segment BD.
Using the trigonometric ratio for sine:
sin(angle) = opposite/hypotenuse
sin(75) = BD/210
Solving for BD:
BD = sin(75) * 210
Step 3: Adding the length of segment BC, which is also 210 meters, to the calculated BD, we get the distance between the separated ends of the sections when the bridge is fully opened.
Distance = BD + BC
Please note that step 2 in both parts uses a trigonometric ratio of either tangent or sine, depending on the calculation being performed. The angles and side lengths provided in the problem are used in the appropriate trigonometric ratios to find the required values.
To solve this problem, we can use trigonometry and the concept of right triangles. Let's break down the problem step by step.
a. Finding the distance from the water to point A on the upper left corner of the right section when the bridge is fully opened:
1. We have a right triangle, where the height of the triangle is the distance from the water to point A, and the base of the triangle is the length of the right section of the bridge (210 meters).
2. The maximum angle of elevation (angle formed between the horizontal line and the line of sight to the top of the bridge) is given as 75 degrees.
3. We can use the trigonometric function tangent (tan) to calculate the height of the triangle. The formula we can use is tan(angle) = opposite/adjacent.
Using the formula: tan(75 degrees) = height/210 meters.
To solve for the height, we can multiply both sides of the equation by 210 meters:
height = tan(75 degrees) * 210 meters.
Using a scientific calculator, we find that tan(75 degrees) is approximately 3.73205. Multiplying this by 210 meters gives us:
height ≈ 3.73205 * 210 meters ≈ 783.33 meters.
Therefore, the distance from the water to point A on the upper left corner of the right section, when the bridge is fully opened, is approximately 783.33 meters.
b. Finding the distance between the separated ends of the sections when the bridge is fully opened:
1. In this case, we have two right triangles formed by the separated ends of the sections when the bridge is fully opened.
2. The angle between the two sections of the bridge is a right angle (90 degrees), as they are perpendicular to each other.
3. The distance between the separated ends of the sections is the hypotenuse of this right triangle.
To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's assume the length of the hypotenuse (distance between the separated ends of the sections) is represented by 'x.'
Using the Pythagorean theorem: x^2 = (length of one section)^2 + (length of the other section)^2
Substituting the given length of each section (210 meters), we have:
x^2 = 210^2 + 210^2
Simplifying the equation:
x^2 = 44100 + 44100
x^2 = 88200
Taking the square root of both sides:
x = √88200
Using a calculator, we find that √88200 is approximately 296.67 meters.
Therefore, the distance between the separated ends of the sections when the bridge is fully opened is approximately 296.67 meters.