A plane flying with a constant speed of 24 km/min passes over a ground radar station at an altitude of 8 km and climbs at an angle of 20 degrees. At what rate, in km/min, is the distance from the plane to the radar station increasing 2 minutes later?

I draw the figure, labeling the distance the plane flies on its path as Flightdistance(FD), d is the distance from radar to plane, 8km is the initial vertical altitude.

FD=time*24km/min

Using law of cosines

d=sqrt(FD^2+8^2-2*8*FD cos 110)
taking the derivative
d'= 1/2*1/sqrt ( ) * (2FD FD'-16cos110*FD')

d' is what you are looking for
FD' given as 24km/min, FD= 24*2=48km

so, it is plug and chug at this point.

To find the rate at which the distance from the plane to the radar station is increasing 2 minutes later, we first need to find the horizontal and vertical components of the velocity of the plane.

Given that the plane is flying with a constant speed of 24 km/min and climbing at an angle of 20 degrees, we can use trigonometry to find the horizontal and vertical components of the velocity.

The horizontal component of the velocity can be found using the equation:

horizontal velocity = speed * cosine(angle)

horizontal velocity = 24 km/min * cosine(20 degrees)

horizontal velocity ≈ 22.48 km/min

The vertical component of the velocity can be found using the equation:

vertical velocity = speed * sine(angle)

vertical velocity = 24 km/min * sine(20 degrees)

vertical velocity ≈ 8.19 km/min

Now, let's use these components to find the rate at which the distance from the plane to the radar station is increasing 2 minutes later.

Since the plane is flying at a constant speed, the horizontal component of the velocity remains the same. Therefore, the rate of change of the horizontal distance will be zero.

The rate at which the distance from the plane to the radar station is increasing is equal to the rate of change of the vertical distance. We need to find the rate of change of the vertical distance 2 minutes later.

Since the vertical velocity is constant, we can use the equation:

rate of change of vertical distance = vertical velocity

rate of change of vertical distance ≈ 8.19 km/min

Thus, the rate at which the distance from the plane to the radar station is increasing 2 minutes later is approximately 8.19 km/min.

To find the rate at which the distance from the plane to the radar station is increasing, we can use the concept of trigonometry.

Let's consider a right-angled triangle, where the base represents the horizontal distance between the plane and the radar station (which we need to find), the height represents the altitude of the plane (8 km), and the hypotenuse represents the distance between the plane and the radar station (which is constant).

We know that the plane is climbing at an angle of 20 degrees, and the speed of the plane is given as 24 km/min. Using trigonometry, we can determine the rate at which the distance between the plane and the radar station is increasing.

First, let's find the vertical component of the plane's velocity:
Vertical component = speed * sin(angle)
Vertical component = 24 km/min * sin(20 degrees)

Next, we can find the horizontal component of the plane's velocity:
Horizontal component = speed * cos(angle)
Horizontal component = 24 km/min * cos(20 degrees)

Now, we have the horizontal and vertical components of the plane's velocity. We can use these values to find the horizontal distance traveled by the plane in two minutes. Since the plane is flying at a constant speed, the horizontal component of velocity remains the same.

Horizontal distance traveled in 2 minutes = Horizontal component * time
Horizontal distance traveled in 2 minutes = (24 km/min * cos(20 degrees)) * 2 min

Finally, to find the rate at which the distance from the plane to the radar station is increasing, we divide the horizontal distance traveled in 2 minutes by the hypotenuse (which is constant):

Rate of increase = Horizontal distance traveled in 2 minutes / Hypotenuse
Rate of increase = [(24 km/min * cos(20 degrees)) * 2 min] / hypotenuse

Now, plug in the given values to find the rate at which the distance from the plane to the radar station is increasing.