A drawbridge is l =180 feet long when stretched across a river. As shown in the

Question

A drawbridge is l =180 feet long when stretched across a river. As shown in the figure below, the two sections of the bridge can be rotated upward through an angle of alpha = 37 degree. (a) If the water level is 35 feet below the dosed bridge, find the distance of between the end of a section and the water level when the bridge is fully open (Round your answer to one decimal place .) (b) Approximately how far apart are the ends of the two sections when the bridge is fully opened, as shown in the figure? (Round your answer to one decimal place.)

Explanation / Answer

Each half of the 180-foot span forms a right triangle when it's opened. One side is the height above its closed position and the hypotenuse is the 90 feet length of the section.

Since angle A is 37 degrees, the sine of 37 degrees is the ratio of the height above the closed position (the opposite side) divided by the 90-foot hypotenuse.
sin(35) = h / 90
h = 90 * sin(37) = 90 * 0.6018 = 54.16 feet
height of the river= add the 35 feet that the bridge is above the river.

The distance between the end of each section and the water is 89.1 feet.


To calculate the distance between the two ends

the two right triangles formed between the bridge segments and the endpoints of the bridge. Since the bridge is raised to 37degrees, the angle between a vertical line drawn through the endpoint of the bridge and the bridge section is 90 - 37 =53 degrees. The hypotenuse is still the 90-foot bridge section. The distance which you need to calculate is from the end of the segment back to the vertical line through the bridge's endpoint. Again the sine function is used because the distance is opposite the 53 degree angle:

sin(53) = d / 90

d = 90 * sin(53) = 90 * 0.7986= 71.87 feet

b) Each section sticks out 71.87 feet from the end of the bridge to which it's attached. Since each side sticks out equally, the gap between the two ends is twice that value subtracted from the total span:

gap = 180 - (2 * 71.87) = 180 - 143.74 = 36.2 feet .

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