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 determine the values of A, B, and w in the general equation p(x) = ACos(wx) + B.
Given conditions:
1. P(0) = 0: The altitude is 0 feet when the plane is at the runway. This means p(0) = 0, so we have 0 = ACos(0) + B. Since Cos(0) = 1, we get A + B = 0.
2. P'(0) = 0: The rate of change of altitude is 0 feet per mile when the plane is at the runway. To find P'(x), we need to differentiate p(x) with respect to x. Differentiating ACos(wx) + B gives us -Awsin(wx). Substituting x = 0, we have 0 = -AwSin(0), which means Aw = 0. Since we want to find the maximum acceleration, we assume w ≠ 0. Hence, A = 0.
3. P(45) = 20000: The altitude is 20000 feet when the plane is 45 miles from the runway. Substituting x = 45 into p(x), we get 20000 = ACos(45w) + B.
4. P'(45) = 0: The rate of change of altitude is 0 feet per mile when the plane is 45 miles from the runway. To find P'(x), we need to differentiate p(x) with respect to x. Differentiating ACos(wx) + B gives us -Awsin(wx). Substituting x = 45, we have 0 = -Awsin(45w).
To find the maximum acceleration, we need to find the maximum value of |Awsin(45w)|. Since the plane is maintaining a constant horizontal velocity of 250 mph, its horizontal distance from the runway is given by x = 250t, where t represents time.
Now, let's solve for A, B, and w:
1. From condition 1, we have A + B = 0.
2. From condition 2, we have Aw = 0. Since w cannot be zero, we have A = 0.
3. From condition 3, we have B = 20000.
4. From condition 4, we have 0 = -Awsin(45w). Since A = 0, we can't determine the value of w from this condition.
Therefore, the function p(x) = 20000Cos(wx) gives the altitude in feet when the plane is x miles from the runway, A = 0, B = 20000, and w can be any real number greater than zero. Unfortunately, we cannot determine the maximum acceleration experienced by the passengers without knowing the value of w.