A small aircraft starts its descent from an altitude of 1 mile, 4 miles west of the runway.
a) Find the cubic f(x) = ax3 + bx2 + cx + d on the interval [-4,0] that describes a smooth glide path for the landing.
b) If the glide path of the plane is described by the function in part a, when would the plane be descending at the most rapid rate?
I need to show step-by-step work for this problem, so please format your answer as such. This is from the chapter "Applications of Differentiation." Thanks! :)
Sorry, that's f(x) = ax^3 + bx^2 + cx + d
get beaned
Sure! Let's break it down step by step:
a) To find the cubic function f(x) = ax^3 + bx^2 + cx + d that describes a smooth glide path for the landing, we need to determine the values of the coefficients a, b, c, and d.
Step 1: Define the conditions for the trajectory. Given that the aircraft starts at an altitude of 1 mile, 4 miles west of the runway, we can use this information to set up the following conditions:
f(-4) = 1 (the altitude at x = -4 is 1 mile)
f(0) = 0 (the altitude at x = 0 is 0 mile)
Step 2: Substitute the x-values into the function f(x) and solve for the coefficients. Let's start with the condition f(-4) = 1:
f(-4) = a(-4)^3 + b(-4)^2 + c(-4) + d = 1
This simplifies to:
-64a + 16b - 4c + d = 1
Now, let's use the second condition, f(0) = 0:
f(0) = a(0)^3 + b(0)^2 + c(0) + d = 0
Simplifying this gives us:
d = 0
Now we have two equations:
-64a + 16b - 4c + d = 1 --> -64a + 16b - 4c = 1
d = 0
Step 3: Solve the two equations simultaneously to find the remaining coefficients. Substituting d = 0 into the first equation, we get:
-64a + 16b - 4c = 1
Now we have two equations with two variables. Solving this system of equations will yield the values of a, b, and c.
There are different methods to solve this system, such as substitution or elimination. You can choose the method you find most comfortable. Once you find the values of a, b, and c, plug them back into the cubic function f(x) = ax^3 + bx^2 + cx + d to obtain the final equation for the glide path.
b) To find the point at which the plane is descending at the most rapid rate, we need to find the critical points of the cubic function. These points occur where the derivative of the function is equal to zero.
Step 1: Differentiate the cubic function f(x) with respect to x. Using the power rule, we have:
f'(x) = 3ax^2 + 2bx + c
Step 2: Set the derivative equal to zero and solve for x to find the critical points:
3ax^2 + 2bx + c = 0
Step 3: Solve the quadratic equation to find the values of x where the plane is descending at the most rapid rate.
You can use the quadratic formula or factoring to solve for x. Once you find the critical points, substitute them back into the original cubic function f(x) to determine at what x-values the plane is descending at the most rapid rate.
Remember, it's essential to follow these steps and perform the necessary calculations to find the cubic function and the critical points accurately.