A plane is flying 12,000 feet horizontally from a tall, vertical cliff. The angle of elevation from the plane to the top of the cliff is 45degrees, while the angle of depression from the plane to a point on the cliff at elevation 8000 feet is 14degrees. Find the height of the cliff

If the plane is at 12000', then the cliff height h is found using

h = 12000 + 4000 sec14° = 16122

b=8944ft

I have this same question. It depends on how the question is read. Is the plane at an unknown hight 12,000 ft from the cliff? Or is the plane an unknown distance from the cliff flying at 12,000 feet? If the second then the cliff is 28,043.1 ft high. If the first AND we assume that the plane is above 8000 ft, then the cliff is 22,991.9 ft high. But what if the plane is flying lower then 8000, then I'm not sure how to read the "angle of depression". (Which might mean it HAS to be higher then 8000 ft).

I don't htink I was much help here. :p

21999 ft

To find the height of the cliff, we can use trigonometry.

First, let's draw a diagram to visualize the problem. Let:
- A be the top of the cliff.
- B be the point on the cliff at elevation 8000 feet.
- C be the plane when it is directly above point B.
- D be the perpendicular from the plane to point B.

Now, we have a right triangle ACD, where:
- AC is the horizontal distance between the plane and the cliff (12,000 feet).
- AD is the height of the cliff (which we want to find).
- Angle DAC is the angle of elevation, which is 45 degrees.
- Angle ACD is the angle between the horizontal (distance AC) and the line connecting C to B, which is the complement of the angle of depression (14 degrees).

Using the given information, we can calculate the height of the cliff.

Step 1: Finding the length of segment CD.
We know that tangent of angle ACD is the ratio of the opposite side (AD) to the adjacent side (AC).
So, tan(ACD) = AD / AC
Since we know the angle and the length of AC,
tan(14) = AD / 12,000
AD = 12,000 * tan(14)

Step 2: Finding the length of segment AB.
Since angle BAC is 90 degrees, the length of AB is equal to the difference between the heights of point A and point B.
AB = altitude of A - altitude of B
AB = 12,000 - 8,000

Step 3: Finding the height of the cliff, AD.
By applying the Pythagorean theorem in triangle ACD, we have:
AD^2 = AC^2 - CD^2
AD^2 = (12,000)^2 - (12,000 * tan(14))^2

Finally, we can calculate the height of the cliff, AD, by taking the square root of the right side of the equation:
AD = √[(12,000)^2 - (12,000 * tan(14))^2]

Performing the calculations will give us the height of the cliff.