An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The problem involves a radical function f ( x ) = a ( x + k ) 1/ n + c .
The domain depends on whether n is even or odd. If n is even, x ≥ − k , and if n is odd, the domain is all real numbers.
The range depends on the sign of a and the value of c .
Assuming a typical radical function where the domain is bounded on one side, the domain is [ − k , ∞ ) and the range is [ c , ∞ ) , so the answer is A .
Explanation
Understanding the Problem We are given a radical function in the form f ( x ) = a ( x + k ) 1/ n + c and asked to determine its domain and range based on the given options. The key is to understand how the parameters a , k , n , and c affect the domain and range.
Analyzing the Function Let's analyze the general form of the function. The domain of the function depends on the value of n . If n is even, the expression inside the radical, x + k , must be non-negative. If n is odd, x + k can be any real number. The range of the function depends on the sign of a and the value of c .
Case 1: n is even If n is even, then x + k ≥ 0 , which implies x ≥ − k . Thus, the domain is [ − k , ∞ ) . If 0"> a > 0 , the range is [ c , ∞ ) . If a < 0 , the range is ( − ∞ , c ] .
Case 2: n is odd If n is odd, then x + k can be any real number, so the domain is ( − ∞ , ∞ ) . The range is also ( − ∞ , ∞ ) regardless of the sign of a .
Comparing with the Options Now, let's compare these possibilities with the given options:
A. The domain is [ − k , ∞ ) , and the range is [ c , ∞ ) . This corresponds to the case where n is even and 0"> a > 0 .
B. The domain is ( − ∞ , k ] , and the range is [ c , ∞ ) . This is not a possible domain for a radical function of the given form. C. The domain is and the range is ange is [ − k , ∞ ) . This option is incomplete and thus cannot be correct. D. The domain is ( − ∞ , ∞ ) , and the range is ( − ∞ , ∞ ) . This corresponds to the case where n is odd.
Making an Assumption Without a graph, we cannot definitively determine whether n is even or odd, or whether a is positive or negative. However, option A is a valid possibility when n is even and 0"> a > 0 , and option D is valid when n is odd. Since the problem states 'the graph shown below', we can assume that the graph is a typical radical function where the domain is bounded on one side. Therefore, we can assume that n is even and 0"> a > 0 .
Final Answer and Conclusion Therefore, the most likely answer is A. The domain is [ − k , ∞ ) , and the range is [ c , ∞ ) .
Examples
Radical functions are used in various real-world applications, such as calculating the period of a pendulum or modeling the velocity of an object in free fall. Understanding the domain and range of these functions is crucial for determining the possible values of the variables involved. For example, if we are modeling the distance an object falls over time using a square root function, the domain would represent the time interval, and the range would represent the distance fallen. Knowing the domain and range helps us interpret the physical meaning of the model and make accurate predictions.
Approximately 2.81 billion billion electrons flow through the electric device in 30 seconds when a current of 15.0 A is delivered. This is calculated using the relationship between current, charge, and time. The total charge is found to be 450 C, and dividing this by the charge of an electron gives the total number of electrons.
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