91Ó°ÊÓ

The Pythagorean theorem is a fundamental concept in geometry that helps us relate the lengths of the sides of a right triangle. This theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In the context of the airplane problem, the distance between the plane and the tower forms the hypotenuse of a right triangle, where the other two sides are the horizontal distance of the plane from the tower and the vertical height above the ground.

• Hypotenuse: The straight-line distance, denoted as \( s \).
• Other two sides: Horizontal distance \( x \) and vertical height \( y \).

The equation becomes \( s^2 = x^2 + y^2 \) which is the starting point to find out how fast the distance \( s \) is changing with respect to time. This setup allows us to use the Pythagorean theorem to form the basis of the related rates problem.

Differentiation is a mathematical process used to find how a function changes over an interval or at a specific point. It plays a crucial role in solving related rates problems like this one. The goal is to determine the rate of change of one quantity with respect to another quantity.
For our problem, we differentiate the equation \( s^2 = x^2 + y^2 \) with respect to time \( t \). This process involves taking derivatives of each term involved:

• Differentiate \( s^2 \) leading to \( 2s \frac{ds}{dt} \).
• Differentiate \( x^2 \) leading to \( 2x \frac{dx}{dt} \).
• Differentiate \( y^2 \) leading to \( 2y \frac{dy}{dt} \).

After differentiation, we obtain the equation \( s \frac{ds}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt} \). This result will help us plug in the known values to solve for the unknown rate \( \frac{ds}{dt} \), which is the changing distance between the plane and tower.

Kinematics is the branch of physics that deals with motion, without considering the forces that cause it. It helps us describe the motion of objects by their speed, velocity, and acceleration over time.
In the exercise, kinematics appears when considering the motion of the airplane:

• The plane's horizontal speed or eastward velocity: \( \frac{dx}{dt} = 640 \) km/hr.
• The vertical climbing speed: \( \frac{dy}{dt} = 10.8 \) km/hr.

These rates of motion are given and must be incorporated into the differentiation step. Understanding these speeds as part of the problem's kinematic perspective allows us to correctly put values into our related rate equation and see how it affects the overall motion concerning the tower.

The concept of rate of change lies at the heart of calculus and is essential for related rates problems. Here, we're focused on how quickly the distance from the plane to the tower changes.
The rate of change for our situation is represented by \( \frac{ds}{dt} \), which we calculate after setting up the differentiated equation. Using given speeds:

• Horizontal rate change \( \frac{dx}{dt} = 640 \) km/hr.
• Vertical rate change \( \frac{dy}{dt} = 10.8 \) km/hr.

We substitute these into the equation \( 7.81 \frac{ds}{dt} = 6 \times 640 + 5 \times 10.8 \). After simplification, we derive the rate of change of the hypotenuse and find that the distance changes at approximately \( 498.34 \) km/hr. This output helps us understand how various changes together alter other measurements dynamically.