An observer is 20 m above the ground floor of a large hotel atrium looking at a glass-enclosed
elevator shaft that is 20 m horizontally from the observer (see figure). The angle of elevation of the elevator is the angle
that the observer's line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator
rises at a rate of 5 m/s, what is the rate of change of the angle of elevation when the elevator is 10 m above the ground?
When the elevator is 40 m above the ground?
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To find the rate of change of the angle of elevation, we need to differentiate the angle with respect to time.
Let's first define our variables:
- h: height of the elevator above the ground (in meters)
- x: horizontal distance from the observer to the elevator (in meters)
- θ: angle of elevation of the elevator
We know that the observer is 20 m above the ground floor and the elevator shaft is 20 m horizontally from the observer. This forms a right-angled triangle with the elevator as the hypotenuse.
Using the Pythagorean theorem, we can find the distance between the observer and the elevator:
x^2 + 20^2 = h^2
Differentiating both sides of the equation with respect to time (t), we get:
2x(dx/dt) = 2h(dh/dt)
Since the observer's line of sight makes an angle of elevation with the horizontal, we can define:
tan(θ) = h/x
Differentiating both sides of this equation with respect to time (t), we get:
sec^2(θ) dθ/dt = (1/x)(dx/dt) - (h/x^2)(dx/dt)
Rearranging the equation and substituting dx/dt = 5 m/s, we have:
dθ/dt = [(1/x)(dx/dt) - (h/x^2)(dx/dt)] / sec^2(θ)
dθ/dt = [(5/x) - (h/x^2)] / sec^2(θ)
Now, we can substitute the given values to find the rate of change of the angle of elevation when the elevator is 10 m above the ground and when it is 40 m above the ground.
For h = 10 m:
Using the equation x^2 + 20^2 = 10^2, we can solve for x:
x^2 + 400 = 100
x^2 = 100 - 400 = -300 (since x is a distance, it cannot be negative, so this solution is not valid)
For h = 40 m:
Using the equation x^2 + 20^2 = 40^2, we can solve for x:
x^2 + 400 = 1600
x^2 = 1600 - 400 = 1200
x = √1200 ≈ 34.64 m
Now, substituting the values into the equation dθ/dt = [(5/x) - (h/x^2)] / sec^2(θ):
When h = 10 m:
dθ/dt = [(5/34.64) - (10/34.64^2)] / sec^2(θ)
When h = 40 m:
dθ/dt = [(5/34.64) - (40/34.64^2)] / sec^2(θ)
Note: The sec^2(θ) term depends on the angle of elevation and is not given in the question. To calculate it, you need to know the specific angle.
Therefore, the rate of change of the angle of elevation depends on the specific angle the observer is looking at.