Question 560304
see the diagram below:
<img src = "http://theo.x10hosting.com/2012/jan211.jpg" alt = "$$$$" />
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