a line from top of cliff to ground just passes over the top of a pole 20m high. the line meet the ground at point 15m from the base of the pole. if it 120m away from this point to the base of the cliff, how high is the cliff
draw a diagram. Usung similar triangles, the height h of the cliff can be found via
h/120 = 20/15
H/120=20/75
H/120=20/75
H=160
Thank you very much
120/15=h/20
=160m
ok
To find the height of the cliff, we can use the concept of similar triangles. We have two triangles in this scenario: the first triangle consists of the cliff, the pole, and the line from the top of the cliff to the ground, and the second triangle consists of the pole, the line from the top of the cliff to the ground, and the line from the base of the pole to the base of the cliff.
Let's label the height of the cliff as 'h' and the distance from the base of the pole to the base of the cliff as 'x'.
In the first triangle, the ratio of the height of the cliff to the distance from the base of the pole to the point where the line meets the ground is the same as the ratio between the height of the pole and the distance from the base of the pole to the point where the line meets the ground. Therefore, we can set up the following proportion:
h / (15 + x) = 20 / 15
To find a relation between 'h' and 'x', we need to establish a connection between 'x' and the given information. We know that the line from the point where it meets the ground to the base of the cliff is 120m long. So, if we subtract 'x' from the total length of 120m, we'll get the remaining distance from the point where the line meets the ground to the base of the cliff. It can be expressed as:
120 - x
Now we can substitute this in the previous equation:
h / (15 + x) = 20 / 15
Simplifying the equation, we get:
15h = 20(15 + x)
Next, let's solve for 'x' by taking it to the other side of the equation:
15h - 20(15) = 20x
15h - 300 = 20x
Now, we need to find the relation between 'x' and 'h' to eliminate 'x' from the equation. We can observe that the remaining distance from the point where the line meets the ground to the base of the cliff, 120 - x, forms another right-angled triangle with the height of the cliff. Therefore, we can set up another proportion:
x / h = (120 - x) / 20
Cross-multiplying, we get:
x * 20 = h * (120 - x)
20x = 120h - hx
Bringing all the 'x' terms to one side of the equation and the 'h' terms to the other side:
hx + 20x = 120h
Factoring out 'x' from the left side:
x(h + 20) = 120h
Now we have two equations with different forms:
15h - 300 = 20x ...(1)
x(h + 20) = 120h ...(2)
To solve for 'h', we can substitute the value of '20x' from equation (1) into equation (2):
x(h + 20) = 120h
(15h - 300)(h + 20) = 120h
Expanding the left side of the equation:
15h^2 + 300h - 6000 = 120h
Bringing all the terms to one side of the equation:
15h^2 + 300h - 120h - 6000 = 0
Simplifying:
15h^2 + 180h - 6000 = 0
Next, let's solve this quadratic equation to find the value(s) of 'h'. We can either factor it or use the quadratic formula, which will give us:
h ≈ 26.39 or h ≈ -40.26
Since the height of the cliff cannot be negative, we can disregard the negative value.
Therefore, the height of the cliff is approximately 26.39 meters.