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The Pythagorean theorem is essential when working with triangles, especially right triangles. It relates the lengths of the sides of a right triangle by stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In mathematical form, it's expressed as:

• \(a^2 + b^2 = c^2\)

For problems involving related rates, like the distance between a horizontally moving motorcycle and a vertically rising balloon, this theorem helps establish important relationships between the distances. By positioning the motorcycle and the balloon's path as perpendicular to each other, we can identify the hypotenuse as the changing distance \(z(t)\) between the two objects. Here, \(x(t)\) and \(y(t)\) are the horizontal and vertical distances, respectively.

• We compute \(z(t)^2 = x(t)^2 + y(t)^2\)
• This sets the stage to apply differentiation to find related rates.

Differentiation is a technique applied in calculus that deals with finding how a function changes at any given point. In related rates problems, we specifically look at how different variables change with respect to time. Here, the distances \(x(t)\), \(y(t)\), and \(z(t)\) change as functions of time.We differentiate with respect to time \(t\) to understand these changes. For our situation, differentiation helps identify:

• \(\frac{d}{dt}[x(t)]\)
• \(\frac{d}{dt}[y(t)]\)
• \(\frac{d}{dt}[z(t)]\)

Using the chain rule, we relate the rate of change of \(z(t)\) to the rates of change of \(x(t)\) and \(y(t)\). This gives us insight into how the overall distance is affected by the movement paths of the motorcycle and the balloon. With differentiation, we extract the variable changes that might not be directly visible but are crucial for solving the problem.
In our equation: \[2z(t) \cdot \frac{d}{dt}[z(t)] = 2x(t) \cdot \frac{d}{dt}[x(t)] + 2y(t) \cdot \frac{d}{dt}[y(t)]\]the derivatives translate into actionable insights for determining unknown rates.

A coordinate system helps us map out the paths of moving objects in mathematical problems. Here, we align our system based on the context:

• The motorcycle starts at the origin \((0,0)\), moving in the positive x-direction.
• The balloon starts 150 ft above ground, moving in the positive y-direction.

This setup provides a visual and mathematical framework for calculating changing distances. By assigning coordinates:

• \(x(t)\) represents the horizontal motion of the motorcycle.
• \(y(t)\) stands for the vertical rise of the balloon.

With such a layout, changes over time can be expressed as equations. It simplifies complex relationships by providing an organized way to apply calculations. In our problem, pinpointing initial positions and paths provides the structure for the differentiation and Pythagorean applications - ultimately aiding the analysis of distance changes.

Rate of change is a concept that explains how one quantity alters relative to another over time. It's particularly emphasized in related rates problems where multiple changing variables are interconnected.In this scenario, we focus on the rate of change of the distance \(z(t)\) between the motorcycle and the balloon. We’re given:

• The horizontal speed of the motorcycle: 58.67 ft/s.
• The ascent rate of the balloon: 10 ft/s.

These rates describe how fast each is moving individually.To find the combined rate at which their mutual distance \(z(t)\) changes, we utilize both given rates in our differential equation.In practical steps, it involves:

• Applying the Pythagorean theorem to determine \(z(t)\) at a specific moment.
• Employing differentiation to compute how \(z(t)\) evolves as \(x(t)\) and \(y(t)\) proceed independently.

Upon such calculation, we found the distance change rate after 10 seconds to be approximately 51.45 ft/s, showcasing how individual speeds translate into a combined dynamic shift.