An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Simplify the expression: 180 \t 2 30 = 2 5400 .
Find the prime factorization of 5400: 5400 = 2 3 \t 3 3 \t 5 2 .
Simplify the square root: 2 5400 = 60 6 .
The equivalent expression is 60 6 .
Explanation
Problem Analysis We are given the expression 180 \t 2 30 and asked to find an equivalent expression from the options: A. 45 \t \t 4 B. 60 \t \t 6 C. 15 \t \t 24 D. 120 \t \t 6
Simplifying the Expression First, let's simplify the given expression. We can rewrite the expression as: 180 \t 2 30 = 2 180 \t 30 = 2 5400
Prime Factorization Now, let's find the prime factorization of 5400: 5400 = 54 \t 100 = 2 \t 27 \t 1 0 2 = 2 \t 3 3 \t ( 2 \t 5 ) 2 = 2 \t 3 3 \t 2 2 \t 5 2 = 2 3 \t 3 3 \t 5 2
Simplifying the Square Root So we have:
2 5400 = 2 2 3 \t 3 3 \t 5 2 = 2 2 2 \t 2 \t 3 2 \t 3 \t 5 2 = 2 \t 2 \t 3 \t 5 2 \t 3 = 2 \t 2 \t 3 \t 5 6 = 60 6
Finding the Equivalent Expression Comparing the simplified expression 60 6 with the given options, we see that option B matches our result.
Final Answer Therefore, the expression equivalent to 180 \t 2 30 is 60 6 .
Examples
Understanding how to simplify radical expressions is useful in various fields, such as physics and engineering, when dealing with calculations involving distances, areas, or volumes. For example, when calculating the length of the diagonal of a rectangle, you might end up with an expression like 180 . Simplifying this expression allows for easier calculations and a better understanding of the actual length. Similarly, in electrical engineering, simplifying radical expressions can help in determining impedance in AC circuits.
The total charge that flows from a device delivering 15.0 A of current for 30 seconds is 450 C . This charge corresponds to approximately 2.81 × 1 0 21 electrons flowing through the device. Therefore, about 2.81 × 1 0 21 electrons pass through the device in that time frame.
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