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Differentiation is a key concept in calculus used to find how a function changes as its input changes. In our problem, the differentiation involves determining how the distance between a balloon and a bicyclist changes over time. Differentiation helps us determine rates of change, which can be seen in this exercise when we calculate the rate at which the distance between the bicyclist and the balloon increases. This is done by taking the derivative of the distance function with respect to time. The derivative, often represented as \(\frac{dd}{dt}\), gives insight into how quickly the distance \(d(t)\) changes as time progresses. Understanding how to differentiate with respect to time is crucial for solving problems involving motion and rates.

A constant rate is an unchanging speed at which an object moves. For example, in this exercise, the balloon ascends at a constant rate of 5 meters per second, while the bicyclist moves at a constant rate of 10 meters per second.Working with constant rates allows us to use simple linear equations to describe the motion. For the balloon, its height after time \(t\) is given by the linear function \(h(t) = 50 + 5t\). Similarly, for the bicyclist, the horizontal distance he covers is \(x(t) = 10t\). These constant rates simplify calculations as each position can be predicted over time using these straightforward linear relationships. This foundation is necessary when using the distance formula to determine the changing distances.

The Distance Formula is a mathematical equation used to determine the distance between two points. In this exercise, it is derived from the Pythagorean theorem. Given the coordinates—vertical distance \(h(t)\) and horizontal distance \(x(t)\)—you can find the total distance \(d(t)\) using the formula:\[d(t) = \sqrt{(x(t))^2 + (h(t))^2}\]Here, \(x(t)\) and \(h(t)\) are the lengths of the sides of a right triangle formed by the balloon and the bicyclist's path. Substituting their expressions (for \(t\) in the formula gives us a functional expression that can be differentiated to find how \(d(t)\) changes over time. This approach is foundational when analyzing real-world problems involving movement.

The chain rule is a technique in calculus used to differentiate composite functions. In the context of the exercise, it allows us to find the rate of change of the distance \(d(t)\) with respect to time, by differentiating a composite function.To apply the chain rule, observe that \(d(t) = \sqrt{(10t)^2 + (50 + 5t)^2}\). The differentiation process involves treating the square root function as an outer function and the squares of \(x(t)\) and \(h(t)\) as inner functions.The chain rule expression simplifies and helps compute the speed at which the distance \(d(t)\) increases as both the balloon and the bicyclist move. It efficiently ties changes in both vertical and horizontal components to find the overall rate of change, offering a more comprehensive understanding of the motion involved.