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 mês, 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?
Steve is wrong the correct answer is
10m = 1/5 rad/s and 40m = 1/8 rad/s
when the elevator is x meters up,
tanθ = (x-20)/20
So,
sec^2θ dθ/dt = 1/10 dx/dt
when x=10, tanθ = -1/2, so sec^2θ = 5/4
5/4 dθ/dt = 1/10 (5)
dθ/dt = 2/5 m/s
Well, well, well, it looks like we've got ourselves an elevator conundrum here. Alright, let's break it down.
When the elevator is 10 m above the ground, we can form a right triangle between the observer, the elevator, and the ground. The vertical leg of the triangle is 10 m (the elevator's height), and the horizontal leg is 20 m (the distance from the observer to the elevator).
Now, we want to find the rate of change of the angle of elevation. In other words, we want to know how fast that angle is changing when the elevator is at a height of 10 m.
To do that, we'll need to use some trigonometry. The tangent of the angle of elevation is equal to the opposite side (10 m) divided by the adjacent side (20 m).
So, tangent(angle) = 10/20.
Now, we'll take the derivative of both sides with respect to time (since we want to find the rate of change with time), let's say t:
sec^2(angle) * d(angle)/dt = (1/2)(10/20) * dt/dt.
Simplifying a bit, we get:
sec^2(angle) * d(angle)/dt = 1/4.
Now, we can plug in the value of the angle when the elevator is 10 m above the ground into the equation to find the rate of change of the angle of elevation. Unfortunately, you didn't provide the angle...
Oh boy, looks like we're missing a vital piece of information. Without knowing the angle, I can't give you a specific answer. Could you provide the angle of elevation at that specific height?
To find the rate of change of the angle of elevation, we need to first find the angle of elevation at the given heights of the elevator.
Let's start by finding the angle of elevation when the elevator is 10 m above the ground.
Here's how you can approach it:
Step 1: Visualize the Problem:
To better understand the problem, let's draw a diagram. Draw a horizontal line to represent the ground floor of the hotel atrium. At a distance of 20 m from the observer, draw a vertical line to represent the elevator shaft. Now, draw a diagonal line connecting the observer to the top of the elevator shaft, forming a right triangle.
Step 2: Identify Known Values:
- The height of the observer above the ground floor: 20 m
- The horizontal distance between the observer and the elevator: 20 m
- The rate at which the elevator rises: 5 m/s
Step 3: Identify the Variables:
Let's call the height of the elevator above the ground "h" and the angle of elevation "θ".
Step 4: Determine the Equation:
Using trigonometry, we know that the tangent of the angle of elevation (θ) is equal to the ratio of the height of the elevator (h) to the horizontal distance (20 m). Therefore, we have the equation: tan(θ) = h/20
Step 5: Find the Angle of Elevation:
When the elevator is 10 m above the ground, the height of the elevator (h) is 10 m. Plugging this into the equation, we get: tan(θ) = 10/20 = 1/2
To find the angle of elevation (θ), we need to take the inverse tangent (arctan) of both sides: θ = arctan(1/2)
Now, you can use a scientific calculator or an online trigonometric calculator to find the value of θ.
Step 6: Calculate the Rate of Change of the Angle of Elevation:
To find the rate of change of the angle of elevation, we need to differentiate the equation with respect to time. However, since we are given the rate at which the elevator rises (5 m/s), we can simply multiply this rate by the derivative of the angle of elevation with respect to height.
So, if we differentiate the equation tan(θ) = h/20 with respect to h, we get: sec^2(θ) * dθ/dh = 1/20
Rearranging the equation, we have: dθ/dh = 1/(20 * sec^2(θ))
Now, plug in the value of θ that you found using the arctan function in Step 5, and also the value of h (10 m) to find the rate of change of the angle of elevation when the elevator is 10 m above the ground.
Similarly, repeat the same steps to find the rate of change of the angle of elevation when the elevator is 40 m above the ground, but use h = 40 m in Step 5.