A line from the top of a cliff to the ground passes just over the top of a pole 5.0 ft high and meets the ground at a point 8.0 ft from the base of the pole. If the point is 79 ft from the base of the cliff, how high is the cliff? Round to the nearest unit.
A) 395 ft
B) 3160 ft
C) 49 ft
D) 5 ft
To solve this problem, we can use the concept of similar triangles.
Let's consider the triangle formed by the top of the cliff, the top of the pole, and the point where the line meets the ground. Let's call this triangle ABC, where A represents the top of the cliff, B represents the top of the pole, and C represents the point where the line meets the ground.
We can also consider another triangle formed by the base of the cliff, the base of the pole, and the point where the line meets the ground. Let's call this triangle ABD, where D represents the base of the cliff.
By considering these two triangles, we can see that triangle ABC is similar to triangle ABD. This means that their corresponding sides are proportional.
The height of the cliff (AD) corresponds to the distance from the base of the pole to the point where the line meets the ground (BD).
We are given that BD is 8.0 ft and the distance from the base of the cliff to the point is 79 ft. We want to find the height of the cliff (AD).
Using the concept of similar triangles, we can set up the following proportion:
AD/BD = AB/BC
We know that BD = 8.0 ft and BC = 5.0 ft (the height of the pole).
Plugging in these values, we get:
AD/8.0 = AB/5.0
To solve for AD (the height of the cliff), we need to find AB.
We are given that the distance from the top of the pole to the top of the cliff is 79 ft (the distance from A to B). We want to find AB.
Using the Pythagorean theorem, we can set up the following equation:
AB^2 = 79^2 - 5^2
Simplifying this equation, we get:
AB^2 = 6241 - 25
AB^2 = 6216
AB = sqrt(6216)
AB ≈ 78.78 ft
Now we can substitute the value of AB back into the proportion:
AD/8.0 = 78.78/5.0
To solve for AD, we can cross multiply:
AD * 5.0 = 78.78 * 8.0
AD ≈ 126.048 ft
Rounding this to the nearest unit, the height of the cliff is approximately 126 ft.
Therefore, the correct answer is D) 126 ft.