SOLUTION: From a certain point you observe that the top of a cliff has an angle of elevation of 45deg. If you move 60 metres directly away from the cliff along the horizontal ground, the ang

Question 560304: From a certain point you observe that the top of a cliff has an angle of elevation of 45deg. If you move 60 metres directly away from the cliff along the horizontal ground, the angle of elevation is now observed to be 30deg. how high is the cliff?
Help please
Found 2 solutions by Theo, mananth:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
see the diagram below:

the height of the cliff is BC which is set to x.
angle C is equal to 90 degrees which makes it perpendicular to the ground.
the distance from the cliff to the base of the 45 degree angle equals DC which is also equal to x because the legs of a 45 degree right triangle are equal.
the length of AD is 60 meters because that's how much you moved back from the 45 degree angle.
the tangent of 45 degrees is equal to BC / DC which is equal to x / x.
the equation is:
tan(45) = x/x
we multiply both sides of this equation to get:
x = x * tan(45)
the tangent of 30 degrees is equal to BC / AC which is equal to x / (x + 60).
we multiply both sides of this equation by (x + 60) to get:
x = (x + 60) * tan(30)
since both expressions are equal to x, we can set them equal to each other to get:
x * tan(45) = (x + 60) * tan(30)
we can simplify this equation to get:
x * tan(45) = x * tan(30) + 60 * tan(30)
if we subtract x * tan(30) from both sides of this equation, then we get:
x * tan(45) - x * tan(30) = 60 * tan(30)
if we factor out the x on the left side of this equation, then we get:
x * (tan(45) - tan(30) = 60 * tan(30)
if we divide both sides of this equation by tan(45) - tan(30), then we get:
x = (60 * tan(30)) / (tan(45) - tan(30))
we use our calculator to solve for x to get:
x = 34.64101615 / .4226497308 which results in:
x = 81.96152423 meters

Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
first observation

45 deg
let distance from clif be x
Tan = opposite side / afjacent side
tan 45 = h/x
x tan 45 = h
similarly
(x+60)*tan 30 =h
tan30*(x+60)=x * tan 45
x tan 30+ 60 tan 60 = x *tan 45
x tan 30 -xtan 45 = -60 * tan 60

x(tan 30-Tan 45)=-60*tan 30
x=-60*tan 30/(tan 30-Tan 45)
x=-34.64/-0.422
x=82
tan 45 = h/82
82*tan 45 = h
height of cliff = 82m

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