A plane flying with a constant speed of 14 km/min passes over a ground radar station at an altitude of 7 km and climbs at an angle of 35 degrees. At what rate, in km/min, is the distance from the plane to the radar station increasing 5 minutes later?
To find the rate at which the distance from the plane to the radar station is increasing, we can use trigonometry and calculus.
Let's denote the distance between the plane and the radar station as d. We are given that the plane climbs at an angle of 35 degrees and flies with a constant speed of 14 km/min. This forms a right triangle between the plane, the radar station, and a point directly below the plane on the ground.
Let's break down the problem step by step:
Step 1: Determine the distance covered by the plane in 5 minutes.
Since the plane is flying at a constant speed of 14 km/min, in 5 minutes, it would cover a distance of 14 km/min * 5 min = 70 km.
Step 2: Determine the height the plane climbs in 5 minutes.
Using trigonometry, we can find the height the plane climbs in 5 minutes by taking the sine of the angle of climb.
Height climbed = 7 km * sin(35 degrees) ≈ 4.038 km.
Step 3: Formulate the equation for the distance from the plane to the radar station.
We can use the Pythagorean theorem to find the distance from the plane to the radar station:
d^2 = (70 km)^2 + (4.038 km)^2
Step 4: Differentiate the equation with respect to time.
Taking the derivative of both sides of the equation with respect to time will give us the rate of change of the distance d with respect to time. Since d is decreasing, we use the negative sign:
-2d * dd/dt = 2(70 km * dd/dt) + 2(4.038 km * dh/dt)
where dd/dt represents the rate at which the distance from the plane to the radar station is changing, and dh/dt represents the rate at which the height is changing.
Step 5: Substitute the given values into the equation.
We are given that the plane is flying at a constant speed and climbing at a constant angle, so dh/dt = 0 km/min.
Substituting the given values, we get:
-2d * dd/dt = 2(70 km * dd/dt) + 2(4.038 km * 0 km/min)
Simplifying further:
-2d * dd/dt = 2(70 km * dd/dt)
-2d * dd/dt - 2(70 km * dd/dt) = 0
Step 6: Solve for dd/dt.
Combining like terms, we get:
dd/dt*(-2d - 2*70 km) = 0
-2d(dd/dt) - 140 km * dd/dt = 0
Now, we can solve for dd/dt:
dd/dt = 0 / (-2d - 140 km)
dd/dt = 0
Therefore, the rate at which the distance from the plane to the radar station is increasing 5 minutes later is 0 km/min.