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Understanding the behavior of current through a resistor can be crucial in electronics. In this concept, the current function is given by the equation: \[i(t)=(60-t)^{2}+(60-t) \sin (\sqrt{t})\] This equation tells us how the current, denoted as \(i(t)\), varies with time \(t\). The function begins with the term \((60-t)^2\), which suggests that the current changes as the time progresses towards 60 seconds. As \(t\) approaches 60, \((60-t)\) becomes zero, causing this term to decrease.
The second term involves a sine function, \((60-t) \sin(\sqrt{t})\), which introduces periodic variations affected by the square root of time. This component may cause the current to fluctuate around the trend set by the quadratic term.
Therefore, to find out how much current is flowing through the resistor at any particular moment, you simply substitute the desired time \(t\) into the current function \(i(t)\).

• This function gives a specific current value at every time \(t\).
• Both components, the quadratic and sinusoidal, play significant roles in defining the behavior of the current.

Ohm's Law is a fundamental principle in electrical circuits and it explains the relationship between voltage (\(V\)), current (\(I\)), and resistance (\(R\)). It is summed up with the equation: \[V = IR\] In our exercise, while we focus primarily on the current \(i(t)\) and resistance \(R\), understanding Ohm's Law helps solidify why knowing these values is essential. According to Ohm’s Law, if you have two of the three values, you can always find the third. In our scenario:

• Ohm’s Law helps us understand why, given a current function \(i(t)\), finding resistance \(R\) is crucial for calculating voltage across the resistor.
• The relationship indicates that any change in current or resistance directly impacts the voltage.

Always consider Ohm's Law when dealing with electric circuits because it provides a logical and mathematical way to relate these critical quantities.

In this exercise, resistance is calculated as a function of the current using the specific equation: \[R = 10i + 2i^{2/3}\] Here, \(R\) denotes the resistance and \(i\) is the current obtained from the resistor current function \(i(t)\). To find the resistance at any given moment, you must first calculate the current using the resistor current function and then substitute that value into the resistance equation.
The resistance function includes two parts:

• The term \(10i\) indicates a linear relationship: as current increases, resistance scales directly with it.
• The second term \(2i^{2/3}\) represents a power-law relation, introducing a more complex dependence of resistance on the current.

Performing these calculations step by step helps ensure accuracy and understanding. First, insert the specific time into \(i(t)\) to get the current. Second, use this current in the resistance equation. This stepwise method makes it straightforward to find the resistance across the resistor at any time.