the line from the top of a cliff to the ground just passes over the top of a pole20m high.the line meets the ground at apoint 15m from the base of the pole.if it is 120m away from this point to the base of the cliff,how high is the cliff?
To solve this problem, we can use similar triangles and the Pythagorean theorem.
Let's label the height of the cliff as 'h'.
Since the line from the top of the cliff to the ground just passes over the top of the pole, the length of this line is: h + 20m.
We are given that this line meets the ground at a point 15m from the base of the pole. So, the length from the top of the pole to this point is: h + 20m + 15m.
We also know that it is 120m from this point to the base of the cliff. Therefore, the length from the top of the pole to the base of the cliff is: h + 20m + 15m + 120m.
Now we can set up a proportion between the similar triangles:
(height of the cliff) / (length from the top of the pole to the base of the cliff) = (height of the pole) / (length from the top of the pole to the ground at the point)
Using the given information, the proportion becomes:
h / (h + 20m + 15m + 120m) = 20m/(h + 20m + 15m)
Now, let's solve this proportion:
h / (h + 155m) = 20m / (h + 35m)
Cross-multiplying, we get:
h * (h + 35m) = 20m * (h + 155m)
Expanding and rearranging, we have:
h^2 + 35hm = 20hm + 3100m^2
Simplifying further:
h^2 + 15hm - 3100m^2 = 0
Now, we can solve this quadratic equation to find the value of 'h', which represents the height of the cliff.