Problem 55 - Single Variable Calculus

Question

Flight Path of an Airplane The path of an airplane on its final approach to landing is described by the equation \(y=f(x)\) with $$ \begin{array}{r} f(x)=4.3403 \times 10^{-10} x^{3}-1.5625 \times 10^{-5} x^{2}+3000 \\ 0 \leq x \leq 24,000 \end{array} $$ where \(x\) and \(y\) are both measured in feet. Estimate the distance traveled by the airplane during the landing approach.

Step-by-Step Solution

Answer

The distance traveled by the airplane during the landing approach can be estimated by calculating the arc length of the given path function, \(f(x)=4.3403\times10^{-10}x^3 - 1.5625\times10^{-5}x^2+3000\), within the interval \([0, 24,000]\). Using the Arc Length formula and simplifying the integral, the path length is represented by: \[ L = \int_0^{24000} \sqrt{1+(1.3021\times10^{-9}x^2 - 3.125\times10^{-5}x)^2} dx \] Since it is not possible to solve this integral analytically, we will use a numerical integration method, such as the Trapezoidal Rule or Simpson's Rule, to approximate the value. By using a calculator or software program like MATLAB or Python, we can estimate the approximate distance L (in feet) that the airplane traveled during the landing approach.

Step 1: Identify the Arc Length Formula

The Arc Length formula is given below, which calculates the length of a curve represented by a function \(y=f(x)\), within the interval \([a, b]\): \[ L=\int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx \] In our case, the function is \(f(x)=4.3403\times10^{-10}x^3 - 1.5625\times10^{-5}x^2+3000\), and the interval is \([0, 24,000]\).

Step 2: Find the Derivative of the Function

We need to find the derivative of the function \(f(x)\) with respect to \(x\). Using the power rule, we get: \[ \frac{dy}{dx} =\frac{d}{dx}(4.3403\times10^{-10}x^3 - 1.5625\times10^{-5}x^2 + 3000) = 3(4.3403)\times10^{-10}x^2 - 2(1.5625)\times10^{-5}x \]

Step 3: Simplify the Derivative

Next, we will simplify the derivative: \[ \frac{dy}{dx} = 1.3021\times10^{-9}x^2 - 3.125\times10^{-5}x\]

Step 4: Insert Derivative into Arc Length Formula and Simplify

We will now insert the derivative we found in Step 3 into the Arc Length formula and simplify: \[ L = \int_0^{24000} \sqrt{1+(1.3021\times10^{-9}x^2 - 3.125\times10^{-5}x)^2} dx \] Now we have the integral that represents the length of the airplane's path during its final approach.

Step 5: Calculate the Length of the Path

We need to solve the integral to find the length of the path. This integral isn't solvable analytically, so we need a numerical method to evaluate it. We will use a numerical integration method, like the Trapezoidal Rule or Simpson's Rule in this case, to get an approximation of the integral value. Note that in practice, you could use a calculator or software like MATLAB or Python to compute the numeric approximation for this integral. Once you find the approximate numerical value (let's call it L) of the integral above, you will have estimated the distance traveled by the airplane during the landing approach. So, the approximate distance the airplane traveled during the landing approach is about L feet (- this will be an actual numeric value you get after calculating the integral numerically).

Key Concepts

Arc Length Formula

Understanding the arc length formula is crucial when we want to find the distance a curve spans between two points. This formula allows us to calculate the length of a path, such as the trajectory of an airplane during its landing approach. It takes the shape of an integral that accounts for the tiny straight segments along the curve – it's like measuring the curve with an infinitely flexible ruler.

To express this mathematically, the arc length formula for a function represented as \( y = f(x) \) from \( a \) to \( b \) is given by:
\[ L=\int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx \]
What this formula essentially does is add up all the infinitesimally small line segments that make up the curve. Therefore, the key to using the formula is being able to find the derivative of the function (\( \frac{dy}{dx} \)) and inputting it into the integral.

Derivative Calculation

The derivative of a function describes how the function value changes as its input changes. It's a fundamental concept in calculus because it gives us the rate of change or the slope of the function at any point. Calculating the derivative often involves applying rules like the power rule, the product rule, or the chain rule.

For the airplane flight path, we calculate the derivative using the power rule. This rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). After applying the power rule to the given function, we obtain:\[ \frac{dy}{dx} = 1.3021\times10^{-9}x^2 - 3.125\times10^{-5}x \]
When it's plugged into the arc length formula, the derivative (which in our case is a quadratic expression) helps determine the curve's steepness and curvature at each point along the x-axis.

Numerical Integration Methods

There are situations, like with the flight path of an airplane, where finding the exact value of an integral is not possible through analytical means. This is when numerical integration methods come to the rescue. These techniques let us approximate the value of definite integrals using algorithms.

Popular numerical methods include:

• The Trapezoidal Rule, which estimates the area under the curve as a sum of trapezoids.
• Simpson's Rule, which uses parabolic arcs to approximate segments of the curve.

Both methods involve partitioning the interval of integration into smaller segments and then calculating the sum of areas of simple geometric shapes that approximate the region under the curve. In practice, you could also use advanced computational tools to help with this task, which are especially useful for functions that are complex or do not have a simple antiderivative.

Calculus Applications

Calculus, particularly the concepts of derivatives and integrals, has a wide array of applications in various fields, from physics to engineering to economics. In the context of our example – determining the flight path distance of an airplane – calculus is used to evaluate the precise length of a curve that cannot be easily measured with straight-edged tools.

Other practical applications include but are not limited to:

• Computing the area of complex shapes or the volume of irregular objects.
• Understanding and describing physical phenomena like motion, electricity, and heat.
• Optimizing functions to find maximum profits or minimum costs in economics.
• Analyzing changing systems over time in population models or infectious disease spread.

In essence, calculus allows us to model and solve problems involving dynamic changes, providing precise calculations where approximations would not suffice.