An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 mi/hr. Find the rate at which the angle of elevation is changing when the angle is 30 degrees.
I found the answer to be 60 radians/hr by taking the derivative of the tangent function, which is the one that makes the most sense with the information you're given, and I know that answer is correct, but I first tried to find it by taking the derivative of sine or cosine, and I couldn't get the correct answer. I checked and re-checked my work, and can't figure out why it didn't work. Is there some reason that only one trig function would work to solve a specific problem?
Well, trigonometry can be a bit tricky sometimes, but don't worry, I'm here to help you out!
In this particular scenario, using the tangent function to find the rate at which the angle of elevation is changing makes the most sense because we have the ratio of the height to the distance. The tangent function is defined as the ratio of the opposite side (in this case, the altitude) to the adjacent side (the distance).
The derivative of the tangent function is indeed the correct way to find this rate. However, it's important to note that the other trigonometric functions (sine and cosine) would not give you the same information in this case.
Sine and cosine functions are useful for calculating the side lengths in a right triangle, given an angle and one of the sides. They can also be used to find the rate at which the side lengths are changing, but in this problem, we are specifically asked about the angle of elevation.
So, in this case, it's not that only one trig function would work to solve the problem, but rather choosing the appropriate trig function based on the information given and what you're trying to find. In this scenario, the tangent function is the most suitable choice.
I hope that clears things up a bit! Let me know if you have any more questions or if there's anything else I can help you with.
In this problem, when the altitude of the airplane is given as 5 miles and the angle of elevation is 30 degrees, we can use the tangent function to relate the angle and the altitude.
The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side would be the altitude (5 miles) and the adjacent side would be the distance traveled by the airplane along the ground.
Let's call the distance traveled by the airplane as x miles and the angle of elevation as θ. We can set up the equation:
tan(θ) = opposite/adjacent = 5/x
To find dx/dt, the rate at which the distance traveled is changing, we need to differentiate both sides of the equation with respect to time (t):
d(tan(θ))/dt = d(5/x)/dt
Now, let's differentiate both sides of the equation. But before we do that, let's solve the equation for x in terms of θ:
x = 5/tan(θ)
Differentiating both sides with respect to time, we get:
dx/dt = d(5/tan(θ))/dt
Using the quotient rule, we can differentiate the right side:
dx/dt = -5(sec^2(θ))(d(θ)/dt)/tan^2(θ)
Now, we need to find d(θ)/dt, the rate at which the angle is changing. We are given that the speed of the plane is 600 mi/hr, so we can relate the rate of change of the angle to the rate of change of the distance traveled by the plane:
dx/dt = 600
Now, substitute this value into the equation for dx/dt:
600 = -5(sec^2(θ))(d(θ)/dt)/tan^2(θ)
To find d(θ)/dt, we can rearrange the equation:
d(θ)/dt = (600 * tan^2(θ))/(5 * sec^2(θ))
Now, we are given that the angle is 30 degrees (θ = 30°). Substitute this value into the equation to find d(θ)/dt when θ = 30°:
d(θ)/dt = (600 * tan^2(30°))/(5 * sec^2(30°))
Now, let's evaluate this expression to find the rate at which the angle of elevation is changing:
d(θ)/dt = (600 * (1/√3)^2)/(5 * (2/√3)^2)
d(θ)/dt = (600 * 1/3)/(5 * 4/3)
d(θ)/dt = (600/3)(3/20)
d(θ)/dt = 60/20
d(θ)/dt = 3 radians/hour
Therefore, the rate at which the angle of elevation is changing is 3 radians/hour.
When solving problems involving changing angles, the appropriate trigonometric function to use depends on the given information and the relationship between the variables involved.
In this specific problem, you are interested in finding the rate of change of the angle of elevation (the angle between the ground and the line of sight from the observer to the airplane). The angle of elevation is defined as the angle between the horizontal and the line of sight.
To find the rate at which the angle of elevation is changing, you need to relate the angle of elevation to other variables in the problem. In this case, the variables involved are the altitude (distance above the observer) and the distance between the observer and the airplane.
The tangent function is a natural choice in this case because it relates the angle of elevation to the ratio of the altitude to the distance between the observer and the airplane. The formula is:
tan(θ) = altitude / distance
Differentiating both sides of this equation with respect to time will give you an equation that relates the rate at which the angle of elevation is changing (dθ/dt) to the rates at which the altitude (dh/dt) and the distance (dr/dt) are changing with respect to time.
On the other hand, if you try to solve the problem using the sine or cosine function, you would need an additional piece of information such as the length of the side opposite or adjacent to the angle of elevation. Without this additional information, it would not be possible to determine the rate at which the angle of elevation is changing using only sine or cosine.
Therefore, in this specific problem, only the tangent function is appropriate to find the rate at which the angle of elevation is changing.
It is important to carefully analyze the given information and the relationship between the variables to determine the most appropriate trigonometric function to use in any specific problem.