By Buildless on November 3rd, 2009
Please solve, and show the steps of how you solved it:
If the Empire State Building and the Sears Tower were situated 1,000 feet apart, the angle of depression from the top of the Sears Tower to the top of the Empire State Building would be 11.53 degrees. The angle of depression to the foot of the Empire State Building would be 55.48 degrees. Find the heights of the buildings.
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Label the top of the Sears Tower as A.
Label the top of the Empire State building as B.
Label the point on the Sears Tower which is level with B as C.
Label the bottom of the Empire State building as D.
Label the bottom of the Sears Tower as E.
Angle ABC is 11.53 degrees. Since the angles within a
triangle are supplementary and angle ACB is 90 degrees,
angle BAC is 180-90-11.53 = 78.47.
Solving for BC using the law of sines:
sin(78.47)/1000 = sin(11.53)/AC
AC * sin(78.47) = sin(11.53)*1000
AC = sin(11.53)*1000 / sin(78.47)
AC = 0.199880994 *1000 / 0.979820181
AC = 203.997629
Angle ADE is 55.8. Using the same logic as used above,
angle DAE is 180-90-55.48 = 34.52
Solving for AE using the law of sines:
sin(34.52)/1000 = sin(55.8)/AE
AE *sin(34.52) = sin(55.8) * 1000
AE = sin(55.8) * 1000 / sin(34.52
AE = 0.827080574 * 1000 / 0.566693877
AE = 1459.4838723
Therefore, the height of the Sears Tower is the length
of side AE, or 1459.48 feet.
The height of the Empire State Building is AE-AC.
AE-AC
1459.4838723 – 203.997629
1255.4862433
Sears Tower: 1459.48
Empire State: 1255.49
To verify, check if computed lengths agree with
pythagorean theorem:
hypotenuse = sqrt(AE^2 + DE^2)
hypotenuse = sqrt(1459.48^2 + 1000^2)
hypotenuse = sqrt(3130081.8704)
hypotenuse = 1769.20374
sin(34.52)/1000 = sin(90)/hypotenuse
sin(34.52)/1000 = sin(90)/1769.20374
0.566693877 / 1000 = 1 / 1769.20374
.000566 = .000565
Within the accuracy of rounding, these are
equal, so the values are consistent.