SOLUTION: 2. Out for a walk in Chicago, Morgan measured the angle of elevation to the distant Sears Tower, and got 3.6 degrees. After walking one km directly toward the building, Morgan foun
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Question 148154: 2. Out for a walk in Chicago, Morgan measured the angle of elevation to the distant Sears Tower, and got 3.6 degrees. After walking one km directly toward the building, Morgan found that the angle of elevation had increased to 4.2 degrees. Use this information to estimate the height of the Sears Tower and how far Morgan is from it after walking toward the building.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
First draw the picture and label the angles and sides:
Now let's find the measures of the other angles. From the drawing, we can see that the angle 4.2 degrees and the unkown angle "a" make up a straight line. So this means that . Solving for "a", we get .
Using the fact that all angles in a triangle add to 180 degrees, this means that (these are the angles from the obtuse triangle). Solving for "b" gives us .
Finally, we know that angles "c" and 4.2 are complementary. So this means that . Isolate "c" and we get
Now add these angle measures to the drawing.
Notice how we have a lot of angle measures but only one side length given. We can use the Law of Sines to find any given length of any triangle as long as we have at least two angles and a side length.
So the Law of Sines formula is:
Since we have two angles and a side, let's use the angle measures 3.6 and 0.6 to find the length of "y"
Plug in , , , and
Simplify.
Multiply both sides by .
Divide both sides by .
Rearrange the equation.
Evaluate the sine functions.
Divide.
So the length of "y" is 5.99616 km
Now let's update the drawing
From the drawing, we can see that the hypotenuse of the second triangle is 5.99616. Using the angle 4.2, this makes the height of the building "x" the opposite side. So let's use the sine function
where "a" is the angle 4.2
Plug in and the given side lengths
Multiply both sides by 5.99616.
Evaluate the sine of 4.2.
Multiply
So the height is . Now to find the length of the base of the second triangle, simply use the cosine function.
where "a" is the angle 4.2
Plug in and the given side lengths
Multiply both sides by 5.99616.
Evaluate the cosine of 4.2.
Multiply
So the distance from the building to the second position is 5.97997 km
---------------------------------
Answer:
So the height of the building is 0.43915 km or 1,440.78084 feet and the distance from the building to Morgan's final position is 5.97997 km or 19,619.3241 feet
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