A military plane maintains an altitude of 15,000 feet over a vast flat desert. It flies at a constant speed on a line that will take it directly over an observer on the ground. At noon, the angle of elevation from the observer’s shoes to the plane is 30 degrees, and the angle of elevation is increasing at a rate of 2.2 degrees per second. Find the speed of the plane at noon. Round off your answer to five significant digits. What is the speed in miles per hour (mph)? Remember that there are 5280 feet in one mile.
The answer is 2303.8 ft/sec or about 1570.8 mph but i don't know how to they arrive at that answer. Please help
As far as I can tell, the time of day makes no difference. So, as usual, draw a diagram.
If the plane is at distance x from the spot directly overhead, its speed is dx/dt, and
tanθ = 15000/x
sec^2θ dθ/dt = -15000/x^2 dx/dt
when θ = 30°, secθ = 2/√3
so, x = 15000/√3 = 8660
2.2°/s = 0.0384 rad/s, so we have
4/3 * 0.0384 = -15000/8660^2 dx/dt
dx/dt = -256 ft/s
I don't like your answer. The angle of elevation is increasing, so the plane is approaching. That means its distance is shrinking, so dx/dt must be negative.
Also, most planes that fly at mach 2 don't do it at 15000 ft.
x = 15000√3 = 25981
4/3 * 0.0384 = -15000/25981^2 dx/dt
dx/dt = -2304 ft/s
My bad.
To find the speed of the plane at noon, we can make use of the relationship between the angle of elevation and the tangent function.
Let's denote the speed of the plane as v (in ft/sec), and the time as t (in seconds) since noon. The angle of elevation θ (in degrees) at time t can be represented as:
θ = 30 + 2.2t
Since the plane is at a constant altitude of 15,000 feet, and we know the tangent of θ is equal to the ratio of the altitude to the horizontal distance covered by the plane, we can set up the following equation:
tan(θ) = 15000 / (vt)
By substituting θ = 30 + 2.2t and rearranging the equation, we get:
tan(30 + 2.2t) = 15000 / (vt)
Simplifying the equation, we obtain:
v = 15000 / (t * tan(30 + 2.2t))
To find the speed of the plane at noon, we need to evaluate the above equation when t = 0:
v = 15000 / (0 * tan(30 + 2.2 * 0))
v = 15000 / (0 * tan(30))
v = 15000 / (0 * 0.5773502691896257)
v = 15000 / 0
This equation results in an indeterminate form (division by zero), which implies that the speed of the plane at noon is undefined.
Therefore, the reported answer of 2303.8 ft/sec or 1570.8 mph is incorrect. Since it is not possible to determine the speed of the plane at noon based on the information provided, there might be an error in the problem statement or solution.