An airplane at a height of 2000 meters is flying horizontally, directly toward an observer on the ground, with a speed of 300 meters per second. How fast is the angle of elevation of the plane changing when this angle is 45 degrees?
At 45 degree elevation angle, airplane altitude h equals horizontal distance from plane to observer, d. (d is the distance from the plane to a point directly above the observer).
A = tan-1 (h/d) is the elevation angle.
dA/dt = d/dh [tan-1 (h/d)]* dh/dt
= V* d/dh [tan-1 (h/d)]
= V*[1/(1 + (h/d)^2)]*(1/d)
When h = d = 2000 m,
dA/dt = (300 m/s)*(1/2)*(1/2000 m)
= 0.075 radians/s
To find how fast the angle of elevation of the plane is changing, we can use trigonometry and differentiate the equation with respect to time. Let's denote the angle of elevation of the plane as θ and the distance between the plane and the observer as x.
We can see that tan(θ) = 2000 / x. Rearranging this equation gives us x = 2000 / tan(θ).
Now, we need to find the rate at which x is changing with respect to time (dx/dt) when θ = 45 degrees. To do this, we differentiate both sides of the equation x = 2000 / tan(θ) implicitly with respect to time (t).
d(x)/dt = d(2000 / tan(θ))/dt.
Using the quotient rule of differentiation, we get:
d(x)/dt = (d(2000)/dt * tan(θ) - 2000 * sec^2(θ) * d(θ)/dt) / (tan^2(θ)).
Now, we need to find d(θ)/dt, the rate at which the angle θ is changing with respect to time (t). We know that the plane is flying horizontally at a speed of 300 meters per second. We can see that the horizontal distance x is changing with the same rate. Therefore, d(x)/dt = 300.
Substituting this value, along with θ = 45 degrees (in radians π/4), into the equation, we get:
300 = (0 - 2000 * (2/√2) * d(θ)/dt) / (1).
Now, we can solve for d(θ)/dt, the rate at which the angle θ is changing with respect to time (t):
d(θ)/dt = (300 * 1) / (-2000 * (2/√2)).
Simplifying the expression further, we get:
d(θ)/dt = -3√2 / 2 radians per second.
Therefore, when the angle of elevation is 45 degrees, the rate at which it is changing is approximately -3√2 / 2 radians per second.