A plane that is flying horizontally at an altitude of 6 kilometers
and a speed of 680 kilometers per hour passes directly over
a radar station. How fast is the distance between the plane
and the radar station increasing when the distance between
the two is 9 kilometers
To find out how fast the distance between the plane and the radar station is increasing, we can use the concept of rates of change and related rates.
Let's assign the following variables:
- The distance between the plane and the radar station: D (in kilometers)
- The time: t (in hours)
We are given:
- The altitude of the plane: 6 kilometers
- The speed of the plane: 680 kilometers per hour
We need to find:
- The rate at which the distance (D) is changing with respect to time (t), i.e., dD/dt
To solve this problem, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (D) is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse (D) represents the distance between the plane and the radar station, and the altitude of the plane (6 kilometers) represents one of the sides of the triangle.
So, using the Pythagorean theorem, we have:
D^2 = (6 kilometers)^2 + (x kilometers)^2
D^2 = 36 + x^2
Now, let's differentiate both sides of the equation with respect to time (t):
2D * dD/dt = 0 + 2x * dx/dt
Since we are interested in finding dD/dt when the distance between the plane and the radar station is 9 kilometers (D = 9), we can substitute these values into the equation above:
2(9) * dD/dt = 0 + 2x * dx/dt
We need to find dx/dt, which represents the rate at which the plane is moving along the horizontal (x) direction. The speed of the plane (680 kilometers per hour) represents the magnitude of dx/dt.
Since the plane is flying horizontally, there is no change in the altitude (6 kilometers). Therefore, dx/dt is equal to the speed of the plane. Thus, dx/dt = 680 kilometers per hour.
Now, we have:
18 * dD/dt = 2x * dx/dt
Substituting the values we know:
18 * dD/dt = 2(9) * (680 kilometers per hour)
Simplifying the equation:
18 * dD/dt = 2 * 9 * 680 kilometers per hour
Solving for dD/dt, which represents the rate at which the distance between the plane and the radar station is increasing:
dD/dt = (2 * 9 * 680 kilometers per hour) / 18
Calculating the value:
dD/dt = 2 * 9 * 680 / 18
Finally, we can calculate the value of dD/dt.