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Differentiation is a mathematical process that enables us to find the rate at which quantities change. When applying differentiation to motion problems, like the one above, we look for how a distance changes over time. This is often expressed as a derivative.

In this scenario, we determine how fast the separation between the car and the train increases. Differentiation helps us understand these changes through a specific formula. When we differentiate the distance formula derived from the Pythagorean theorem, we can calculate this rate of separation. It's crucial to apply appropriate calculus techniques, specifically implicit differentiation, to find the derivative with respect to time.

Implicit differentiation is used when we want to find a derivative where the dependent and independent variables are not explicitly separated. This technique is especially handy in our problem due to how the distances interact within the Pythagorean theorem. As seen here, differentiation offers a clear look at how two objects separate over time in complex motion scenarios.

The Pythagorean theorem is a fundamental principle in geometry, allowing us to find the length of sides in a right triangle. This theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides equals the square of the longest side, called the hypotenuse. Specifically, it is expressed as: \[c^2 = a^2 + b^2\]
The problem involves the paths of the automobile and the train arranged at a right angle; thus this theorem is instrumental in calculating how their positions relate.

In our problem, the automobile and the train form the two shorter sides of a right triangle, while the distance between them after a certain time becomes the hypotenuse. We begin by calculating their respective distances traveled after 10 seconds and use these as the two shorter sides in the equation. This gives us: \[d = \sqrt{660^2 + 100^2}\]
This highlights the timeless usefulness of the Pythagorean theorem in solving problems involving right-angle considerations.

Distance calculation here involves adding together a vertical and horizontal displacement. After 10 seconds, both the car and the train have covered a specific distance. The automobile moves horizontally, covering 660 feet, and the train moves 880 feet along a perpendicular path.

To find out how far the automobile and train are from each other horizontally and vertically, and effectively determine the separation, we compile these displacements.

• Automobile: 660 feet from the starting point.
• Train: 880 feet from its starting point.
• Vertical separation: Always 100 feet.

Using these distances, we calculate the resultant horizontal distance from the Pythagorean theorem. Thus, we understand how the movements translate into the total separation, which is key to reaching our ultimate question of the rate at which they separate.

Velocity conversion is a necessary step to solve physics problems where speed is given in inconsistent units. Often, velocity is expressed in miles per hour (mph), which may need conversion to feet per second (fps) to match other calculations. This is due to the different unit systems involved in physics summaries.

In our case, the original speeds were given in miles per hour: the automobile moves at 45 mph, and the train at 60 mph. To convert these speeds to feet per second:

• Conversion factor: 1 mile = 5280 feet & 1 hour = 3600 seconds.
• Automobile: \(45 \text{ mph} = 45 \times \frac{5280}{3600} = 66 \text{ fps}\).
• Train: \(60 \text{ mph} = 60 \times \frac{5280}{3600} = 88 \text{ fps}\).

Converting to feet per second ensures consistency in our calculations moving forward. It allows for seamless application of mathematical formulas, like those used in the differentiation and Pythagorean theorem calculations.