A surveyor made two sections of the railroad bridge, both at 210 meters in length. Suppose that the maximum 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, what is 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 section.
A figure needs to be provided to answer this question. From where to where is the elevation angle being measured?
Haiwh
To solve these problems, we can use trigonometric functions and basic geometry concepts. Let's break it down step by step:
a. To find the distance from the water to point A on the upper left corner of the right section, we can use trigonometry. From the given information, we know that the maximum elevation of each section is 75 degrees. Let's denote the distance from the water to point A as d.
First, we need to find the vertical height above the water for each section. Given that the water level is normally 13 meters below the bridge, the vertical height of each section can be calculated as follows:
Vertical height = 210 meters + 13 meters = 223 meters
Now, we can use trigonometry to find the distance from the water to point A on the upper left corner of the right section. We can create a right triangle with the vertical height as the opposite side and the distance we want to find (d) as the adjacent side.
Using the tangent function, we can write:
tan(75 degrees) = opposite/adjacent
tan(75 degrees) = 223/d
To solve for d, we rearrange the equation:
d = 223 / tan(75 degrees)
Using a calculator, we find:
d ≈ 64.52 meters
Therefore, the distance from the water to point A on the upper left corner of the right section is approximately 64.52 meters.
b. To find the distance between the separated ends of the section when the bridge is fully opened, we need to consider the geometry of the bridge.
When the bridge is fully opened, the two sections form a straight line. The distance between the separated ends can be calculated by finding the hypotenuse of a right triangle formed by connecting the upper left corner of the left section and the upper right corner of the right section.
Using the Pythagorean theorem, we can write:
Distance between ends = √(210 meters)^2 + (2 * 223 meters)^2
Distance between ends = √(44100 meters^2 + 99692 meters^2)
Distance between ends ≈ √(14379284 meters^2)
Distance between ends ≈ 379.47 meters
Therefore, when the bridge is fully opened, the distance between the separated ends of the section is approximately 379.47 meters.
To solve this problem, we can use trigonometry, specifically the sine and cosine functions, to relate the angles and lengths involved. Let's break it down into two parts:
a. To find the distance from the water level to point A on the upper left corner of the right section, we can use the sine function. By drawing a right triangle with the vertical distance from the water level to point A as the opposite side, and the length of the section as the hypotenuse, we can use the angle of elevation (75 degrees) to find the answer.
First, let's find the length of the adjacent side of the right triangle. We can use the cosine function:
cos(75 degrees) = adjacent side / length of the section
adjacent side = cos(75 degrees) * length of the section
Then, we can find the distance from the water level to point A using the sine function:
sin(75 degrees) = opposite side / length of the section
opposite side = sin(75 degrees) * length of the section
By substituting the length of the section (210 meters) into the equations, we can calculate the values:
adjacent side = cos(75 degrees) * 210 meters
opposite side = sin(75 degrees) * 210 meters
b. To find the distance between the separated ends of the section when the bridge is fully opened, we can use the sine function again. By drawing a right triangle with the vertical distance between the separated ends of the section as the opposite side, and the length of the section as the hypotenuse, we can use the angle of elevation (75 degrees) to find the answer.
Similarly, we can use the sine function:
sin(75 degrees) = opposite side / length of the section
opposite side = sin(75 degrees) * length of the section
By substituting the length of the section (210 meters) into the equation, we can calculate the value of the vertical distance between the separated ends.
Note: Make sure to convert the angles to radians if your calculator uses radians instead of degrees.