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From a point on a line from the base of the Washington Monument, the angle of elevation to the top of the monument is 42.0°. From a point 100 feet away from on the same line, the angle to the top is 37.8°. Find the height, to the nearest foot, of the Washington Monument.

Mathisfornerds  May 4, 2018
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Let the height of the monument  = h

Let the distance from the base of the monument to a point directly under the apex  = x

 

So   we have these equations

 

tan(42)  =  h / x     rearrange as

 

x  =   h / tan (42)      (1)

 

We also have

 

tan (37.8)  =  h / (100 + x)

 

(100 + x) tan (37.8)   = h

 

100*tan(37.8)  + x *tan (37.8)  =  h     (2)

 

Sub  (1)  into (2)

 

100* tan(37.8)  + (h /tan(42)* tan (37.8)  = h

 

100* tan(37.8)  =  h  -  h [ tan (37.8) / tan(42) ]

 

100* tan (37.8)  =  h  [  1  - tan(37.8)/tan(42) ]

 

h  =     [ 100 * tan (37.8) ]  /  [  1 - tan(37.8)/tan(42) ] ≈  560 ft

 

 

cool cool cool

CPhill  May 4, 2018

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