A plane at an altitude of 12,000 feet finds the angle of depression for a building ahead of the plane to be 10.4 degrees. With the building still straight ahead, two minutes later the angle of depression is 25.6 degrees. Find of the plane relative to the ground in feet per minute.

draw a diagram. At first sighting, the distance d1 is

d1/12000 = tan 10.4°

The second distance d2 is

d2/12000 = tan 25.6°

Since speed is distance/time,

s = (d1-d2 ft)/(2 min)

That will be in ft/min.

To find the rate at which the plane is descending relative to the ground, we can use trigonometry.

Let's denote:
- the angle between the flat ground and the line connecting the plane to the building as θ1 = 10.4 degrees,
- the angle between the flat ground and the line connecting the plane to the building after two minutes as θ2 = 25.6 degrees,
- the vertical distance descended by the plane during the two minutes as d in feet,
- the horizontal distance between the plane and the building as x in feet.

Using trigonometry, we can establish two equations:

Equation 1: tan(θ1) = 12,000 / x
Equation 2: tan(θ2) = (12,000 - d) / x

Let's solve these equations simultaneously to find the value of d.

Rearranging Equation 1 to solve for x:
x = 12,000 / tan(θ1)

Substituting x into Equation 2:
tan(θ2) = (12,000 - d) / (12,000 / tan(θ1))

Rearranging Equation 2 to solve for d:
d = 12,000 - (12,000 / tan(θ1)) * tan(θ2)

Now we can calculate the value of d using the given values of θ1 and θ2:

d = 12,000 - (12,000 / tan(10.4)) * tan(25.6)

Using a calculator, calculate the value of d.

d ≈ 4,155.07 feet

Therefore, the plane descended by approximately 4,155.07 feet in two minutes relative to the ground.

To find the rate at which the plane is descending relative to the ground in feet per minute, we divide this distance by the time taken:

Rate of descent = d / 2

Rate of descent ≈ 4,155.07 feet / 2 minutes

Rate of descent ≈ 2,077.54 feet per minute

Therefore, the plane is descending at a rate of approximately 2,077.54 feet per minute relative to the ground.

To determine the rate of descent of the plane relative to the ground in feet per minute, we need to calculate the vertical distance the plane descends in two minutes.

Let's break down the problem step by step:

1. Draw a diagram: Draw a side-view diagram of the situation. Represent the plane above the ground level, the building, and the angles of depression.

2. Identify the given information:
- Altitude of the plane: 12,000 feet
- Angle of depression initially: 10.4 degrees
- Angle of depression after two minutes: 25.6 degrees

3. Find the vertical distances:
- We need to find the vertical distance the plane descends in two minutes. Let's call this distance "d."

- Use trigonometry to calculate these distances. In a right triangle, the tangent function relates the angle of depression to the opposite side (vertical distance) over the adjacent side (horizontal distance).

- For the initial angle of depression:
tan(10.4 degrees) = vertical distance (d) / horizontal distance (x)
Solve for d, the vertical distance.

- For the angle of depression after two minutes:
tan(25.6 degrees) = vertical distance (d) / horizontal distance (2x)
Solve for d using the horizontal distance as 2 times the earlier distance (2x).

4. Calculate the vertical distance:
- Plug in the values into the trigonometric equation for each angle of depression and solve for the vertical distances (d).

5. Determine the rate of descent:
- Divide the total vertical distance (d) covered in two minutes by 2 to determine the rate of descent per minute.
- Since this is the vertical distance covered by the plane relative to the ground, consider negative sign if vertical distance is descended.

Performing the calculations:

1. For the initial angle of depression:
tan(10.4 degrees) = d / x
Rearranging, d = x * tan(10.4 degrees)

2. For the angle of depression after two minutes:
tan(25.6 degrees) = d / (2x)
Rearranging, d = 2x * tan(25.6 degrees)

3. Calculate the values by substituting the known angle and solving for "d":
For the initial angle:
d = x * tan(10.4 degrees)

For the angle after two minutes:
d = 2x * tan(25.6 degrees)

4. Find the difference in height after two minutes:
The difference in height is the second distance minus the first distance.
vertical distance in 2 minutes = (2x * tan(25.6 degrees)) - (x * tan(10.4 degrees))

5. Calculate the rate of descent per minute:
Divide the difference in height by 2 to get the rate per minute.
rate per minute = [(2x * tan(25.6 degrees)) - (x * tan(10.4 degrees))] / 2

Now you can substitute the given angle values and altitude of the plane to find the rate of descent per minute.