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The Distance Formula is a key concept in mathematics used to determine the distance between two points on the coordinate plane. The formula is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This equation is derived from the Pythagorean theorem. By measuring the horizontal and vertical distances between two points and treating them as the legs of a right triangle, we can use the Pythagorean theorem to find the hypotenuse, which in this case represents the distance between the points.
In the problem at hand, we're determining the dynamic distance between two moving ships over time as they traverse through a coordinate plane.Utilizing the distance formula helps us establish a mathematical relationship for the distance which can then be differentiated to investigate how the distance changes with time.

Differentiation is a calculus technique used for finding the rate at which a quantity changes with respect to another quantity. In this problem, we differentiate the distance between two moving ships with respect to time to find out how the distance changes over time.The derivative tells us the instant rate of change or the velocity at which one ship moves relative to the other. By applying differentiation to the distance formula, expressed as a function of time, we can determine exactly how fast the distance between the ships is increasing at any moment.
In this exercise, after forming the distance equation using the distance formula, we differentiate it with respect to time to find \( \frac{dd}{dt} \), which provides the rate of change in the distance between the ships.

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and shapes according to their coordinates, defined by an x (horizontal) and y (vertical) axis. In this context, it represents a grid upon which the paths of two ships are described.By using the coordinate plane, we can clearly define the starting positions of both ships: one at \((0,4)\) and the other at \((-3,0)\). These positions serve as initial points that help set up the equations used to describe the ships' movement over time.
Since the ships move along predictable paths, the coordinate plane helps visualize and calculate how their positions change, allowing us to apply the distance formula effectively.

Velocity in physics refers to the speed of an object in a specific direction. In this problem, each ship has a distinct velocity impacting how their respective positions change over time.For the ship traveling north, the position changes at a velocity of 5 miles per hour. This gives us its new position as \((0, 4 + 5t)\). Similarly, the ship traveling east moves at 4 miles per hour, described by its position \((-3 + 4t, 0)\).
These velocities enable us to model each ship's movement over time. Later, by evaluating how the velocities affect the distance function through differentiation, we can understand how the relative motion of the ships influences the rate at which their distance changes, which is the primary focus of this problem.