SOLUTION: A bridge is going to be built across a canyon from point A to point B. The bearing from A to B is S 28o W. Point C is 87 miles from point A. From point C, the bearing to poin
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-> SOLUTION: A bridge is going to be built across a canyon from point A to point B. The bearing from A to B is S 28o W. Point C is 87 miles from point A. From point C, the bearing to poin
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Question 1120730: A bridge is going to be built across a canyon from point A to point B. The bearing from A to B is S 28o W. Point C is 87 miles from point A. From point C, the bearing to point A is S 84o E and to point B is S 32o E. Find the required length of the bridge.
Round your answer to two decmial places.
You can put this solution on YOUR website! the key here appears to be to get the angles right.
once you do that, you can use the law of sines to find the required distances.
here's my diagram.
you are given that point B is south 28 degrees west from point A.
that would be angle BAE in the diagram.
AE is a perpendicular line dropped from point A.
you are given that point C is 87 miles from point A.
that would be line segment CA in the diagram.
you are given that point A is south 84 degrees east from point C.
that would be angle ACD in the diagram.
CD is a perpendicular line dropped from point C.
you are given that point B is south 32 degrees east from point C.
that would be angle BCD in the diagram.
CD is a perpendicular line dropped from point C.
line segment DE is a horizontal line drawn from point D through point B extending to point E.
angle CDB and angle AEB are both right angles because they are the intersection of horizontal and vertical lines.
that makes angle CDB and AEB both equal to 90 degrees.
angle ABE is equal to 62 degrees because it is part of triangle ABE and the sum of the angles of a triangle is equal to 180 degrees.
similarly, angle CBD is equal to 58 degrees.
angle CBA is equal to 60 degrees because it is the third angle that makes up the 180 degree angle DBE.
angle DBE is equal to angle CBD plus CBA plus ABE.
angle ACB is equal to 52 degrees because it is equal to angle ACD minus angle BCD.
the sum of angles BCD and ACB is equal to 32 + 52 = 84 degrees which is angle ACD.
looks like all the angles are take care of.
you have triangle ABC with side AC = 87.
the other two sides of the triangle can be found using the law of sines.
you get:
AB / sin(52) = 87 / sin(60).
solve for AB to get AB = 87 * sin(52) / sin(60).
CB / sin(68) = 87 / sin(60).
solve for CB to get CB = 87 * sin(68) / sin(60).
you will get AB = 79.16273041 and CB = 93.14391356.
you were asked to find the length of the bridge.
that would be the line segment AB = 79.16273041 miles = 79.16 miles rounded to two decimal places.
here's a reference on bearing that might be helpful.
