Please answer.
An airplane at an altitude of 3000m, flying horizontally at 300km/hr, passes directly over an observer. Find the rate at which it is approaching the observer when it is 5000m away.
Make your sketch, it should be a right-angled triangle
I will assume that the "when it is 5000" refers to the hypotenuse. Make the necessary changes if otherwise.
let the horizontal be 300t, where t is in hrs
the vertical is fixed at 3000
let the hypotenuse be h
when h = 5000
5000^2 =
h^2 = 3000^2 + (300t)^2
when h = 5000
5000^2 = 3000^2 + (300t)^2
300t = 4000
t = 40/3
2h dh/dt = 0 +2(300t)(300)
dh/dt = 90000t/h
= 90000(40/3) / 5000
= 240 km/h
To find the rate at which the airplane is approaching the observer, we can use the concepts of similar triangles and differentiation.
Let's break down the problem into different parts. First, let's draw a diagram to help visualize the situation:
A (airplane)
|\
| \
h | \ r
| \
| \
|_____\
O (observer)
Here, A represents the airplane, O represents the observer, and h is the altitude of the airplane. The observer and the airplane create a right triangle, where r is the horizontal distance between them.
Now, we can use the concept of similar triangles to find a relation between r and h. Since the airplane is flying horizontally, the triangle OAB (where B is the foot of the perpendicular) is similar to triangle OHB. Therefore, we have the following ratio:
r / h = OB / HB
Let's differentiate this equation with respect to time:
d(r/h)/dt = d(OB/HB)/dt
The left side of the equation represents the rate at which the airplane is approaching the observer, which we want to find. The right side of the equation represents the derivative of the ratio between the distances, which can be evaluated using the chain rule:
d(r)/dt * (1/h) - r/h^2 * d(h)/dt = -dB/dt / HB
We are given the airplane's altitude, h = 3000m, and its speed, 300km/hr. We want to find the rate at which the airplane is approaching the observer when it is 5000m away, r = 5000m.
To find dB/dt, we need to consider that the speed of the airplane is given as 300km/hr. Since r is the distance traveled by the airplane in a given time, the rate of change of r with respect to time is equal to the speed of the airplane:
d(r)/dt = 300 km/hr
To find d(h)/dt, we need to consider that the altitude of the airplane is decreasing at a constant rate, as the airplane is flying horizontally. Therefore, d(h)/dt is the rate at which the altitude is decreasing. We are not given this information, so we cannot find the exact value.
However, if we assume that the rate of change of altitude, d(h)/dt, is constant, we can proceed with the calculation. Let's call this constant k:
d(h)/dt = k
Now, substituting all the given and assumed values into the equation and solving for the rate at which the airplane is approaching the observer (d(r)/dt), we get:
300 km/hr * (1/3000m) - 5000m / (3000m^2) * k = -dB/dt / HB
Simplifying further,
d(r)/dt = (300 km/hr) / (3000m) - 5000m / (3000m^2) * k
You can plug in the assumed value of k (if provided) and calculate the rate at which the airplane is approaching the observer. Keep in mind that the rate of change of altitude, d(h)/dt, affects the final answer.