91Ó°ÊÓ

The Pythagorean theorem is a fundamental principle in geometry, especially useful in problems involving right triangles. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This can be written as:

• \( c^2 = a^2 + b^2 \)

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
In the problem of the ships, this theorem helps us calculate the changing distance between the ships. They move perpendicular to each other: ship A southwards and ship B northwards, while starting with a fixed east-west distance between them.
We use the theorem to determine the diagonal distance or the hypotenuse, which represents the distance between the two ships. Calculating this at any point in time involves substituting the values into the theorem to find the distance \( D(t) \).

Differentiation is a powerful mathematical tool for finding how a function changes at any given point, often referred to as the function's derivative.
In calculus, differentiation is used to find the rate at which one quantity changes with respect to another.
In the ships problem, we utilize differentiation to understand how quickly the distance between the ships changes over time. By using the position functions for each ship to form a single distance function, we differentiate this function with respect to time.
Using the chain rule, which is essential in differentiation when dealing with composite functions, helps us solve for \( \frac{dD}{dt} \). This derivative tells us the rate at which the distance between the ships changes as both continue on their paths.

• The chain rule involves taking the derivative of the outer function and multiplying it by the derivative of the inner function, providing a pathway to solve more complex differentiation problems.

The distance formula helps calculate the distance between two points in a coordinate plane. It's particularly handy in physics and engineering, where knowing precise distances is crucial.
The formula is defined as:

• \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.
In the given scenario with the ships, the distance initially is the straight-line connection from ship A to ship B.
The distance formula helps break down the components: the east-west distance is constant at 100 km, while the north-south distance changes over time due to the ships' movements.
Ultimately, we apply this formula in conjunction with the time-dependent changes in directional distances to find the real-time position of the ships relative to each other, then calculate the rate of that changing distance.