if the helicopter is moving forward at 30 mph and drops 400 feet in the elevation while the vehicle is moving 48 mph, how does the angle of depression change after 10 minutes?
So the helicopter drops 400 feet. How high was it when it started its descent?
what is "the vehicle"? Is it a car driving directly underneath the chopper?
Did we start at t=0 when the chopper was directly above "the vehicle"?
Which angle of depression? chopper-to-ground? chopper-to-fixed point on ground? chopper-to-moving "vehicle"?
Woefully incomplete conditions!
However, assuming the chopper starts at (0,h) and the vehicle starts at (0,0),
after t minutes the chopper and vehicle are at
(2640t,h - 5/33 t) and
(4224t,0)
So, the line between them has slope
(h-5/33 t)/(1584t)
Now you have the angle
tanθ = (h-5/33 t)/(1584t)
so you can figure dθ/dt at t=10.
To determine how the angle of depression changes after 10 minutes, we need to consider the motion of the helicopter and the vehicle separately.
Let's break down the problem step by step:
1. Calculate the distance traveled by the helicopter in 10 minutes:
Distance = Speed × Time
Given that the helicopter is moving forward at 30 mph, we convert the time to hours (10 minutes = 10/60 = 1/6 hours):
Distance = 30 mph × (1/6) hour
2. Calculate the change in elevation:
Given that the helicopter drops 400 feet in elevation, we have:
Elevation change = -400 feet (negative as it is dropped)
Now, let's calculate the angle of depression:
3. Determine the horizontal distance covered by the helicopter in 10 minutes:
Horizontal distance = Distance traveled × sin(angle)
The angle of depression is the angle between the line of sight from the observer to the object (helicopter) and the horizontal plane. So, we can use the sin function to find the horizontal distance.
sin(angle) = Elevation change / Distance traveled
Horizontal distance = Distance traveled × sin(angle)
4. Substitute the values into the equation:
Horizontal distance = (30 mph × (1/6) hour) × (Elevation change / Distance traveled)
Simplifying the equation, we find the horizontal distance covered by the helicopter.
As for how the angle of depression changes, it depends on the horizontal distance covered. If the helicopter is moving forward at a constant speed and the angle of depression is defined by the ratio of the elevation change to the horizontal distance, then the angle of depression will remain constant.
However, if the helicopter changes its speed or direction, it will affect the angle of depression. Please provide more context if necessary.