A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 6 mi away from the station. (Round to the nearest whole number.)
Use Pythagoras theorem.
x=horizontal distance from station
L(x)=oblique distance from station
L(x)=√(480²+x²)
dx/dt = horizontal speed = 480 mph
Find dL(x)/dx by differentiation.
Then
dL(x)/dt
= dL(x)/dx * dx/dt
that makes no sense
To find the rate at which the distance from the plane to the station is increasing, we can use the concept of related rates.
Let's denote the distance from the plane to the radar station as "d" and the time as "t". We are given that the plane is flying horizontally at a speed of 480 mi/h, so the rate at which the plane is moving horizontally is 480 mi/h.
We are asked to find the rate at which the distance is increasing when the plane is 6 miles away from the station. In other words, we need to find the rate of change of d with respect to t when d = 6.
Since the plane is flying horizontally and passing directly over the radar station, we can consider the path of the plane as a right triangle. The altitude of the plane (3 mi) forms one side of the triangle, and the distance (d) forms the hypotenuse of the triangle.
Using the Pythagorean theorem, we can relate the altitude, the distance, and the other side of the triangle:
d^2 = (altidute)^2 + (horizontal distance)^2
d^2 = 3^2 + (horizontal distance)^2
d^2 = 9 + (horizontal distance)^2
Differentiating both sides of the equation with respect to time (t), we get:
2d * dd/dt = 0 + 2(horizontal distance) * (d(horizontal distance)/dt)
2d * dd/dt = 2(horizontal distance) * (d(horizontal distance)/dt)
Now, we substitute the given values into the equation:
2(6) * dd/dt = 2(6) * (480)
12 * dd/dt = 12 * 480
12 * dd/dt = 5760
Next, we solve for dd/dt, which represents the rate at which the distance is increasing:
dd/dt = 5760 / 12
dd/dt = 480
Therefore, the rate at which the distance from the plane to the station is increasing when it is 6 miles away from the station is 480 mi/h.