Flight Path. In this problem you will use a cosine function to model the flight path of a plane that begins its descent from 20000 ft when it is 45 miles from the airport. Find the fuction p(x)= ACos(wx)+B that that gives the altitude in ft when the plane is x miles from the runway. your function must satisfy P(0)=0, P'(0)=0 P(45)=20000 P'(45)=0. if the plane is maintaining a constant horizontal velocity of 250mph, what is the maximum acceleration experienced by the passengers?
To find the function P(x) that gives the altitude in feet when the plane is x miles from the runway, we need to apply the given conditions and use the properties of a cosine function.
The general equation for a cosine function is: f(x) = A * cos(wx) + B, where A is the amplitude, w is the frequency, and B is the vertical shift.
Given conditions:
1. P(0) = 0: The altitude at 0 miles from the runway is 0 feet.
2. P'(0) = 0: The derivative of P(x) at x = 0 is 0, which means the plane is not changing altitude at the starting point.
3. P(45) = 20000: The altitude at 45 miles from the runway is 20000 feet.
4. P'(45) = 0: The derivative of P(x) at x = 45 is 0, which means the plane is not changing altitude at 45 miles from the runway.
From condition (1), we have P(0) = A * cos(w*0) + B = A + B = 0.
This implies A = -B.
To find w (frequency), we use the fact that the plane maintains a constant horizontal velocity of 250 mph. The distance traveled x is related to time t by the equation x = vt, where v is the velocity. Since the plane is traveling at 250 mph, we have x = 250t. At t = 1 hour, x = 45 miles. Therefore, 250 * t = 45, which gives t = 45/250 = 9/50 hours.
The frequency (w) of the cosine function is determined by the formula w = 2π / T, where T is the period. In this case, the period T is the time it takes for the plane to travel 45 miles, which is 9/50 hours.
Now, using condition (3), we have P(45) = A * cos(w * 45) + B = -B * cos(w * 45) + B = 20000.
Finally, using condition (4), we have P'(45) = -A * w * sin(w * 45) = 0.
Now we have two equations:
-A * cos(w * 45) + B = 20000
-A * w * sin(w * 45) = 0
Substituting A = -B from the first equation, we get:
B * cos(w * 45) + B = 20000
-B * w * sin(w * 45) = 0
To solve these equations, we need to use numerical methods such as Newton's method or a graphing calculator that supports solving equations. This will give you the specific values of A, B, and w.
Once you have determined the function P(x) = ACos(wx) + B that satisfies all the given conditions, to find the maximum acceleration experienced by the passengers, you need to find the second derivative of P(x) and evaluate it at the point where the first derivative is zero (P'(45) = 0). The value of the second derivative will give you the maximum acceleration.