An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The calculation shows that approximately 2.81 × 1 0 21 electrons flow through the device that delivers a current of 15.0 A for 30 seconds. This is done by first calculating the total charge in coulombs and then determining how many electrons make up that charge. The charge of a single electron is used to find the total number of electrons.
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Determine the elements of set A based on the given conditions: A = { 1 , 3 , 5 } .
Calculate the cardinality of set A: n ( A ) = 3 .
Identify the prime numbers within the universal set U to define set C: C = { 2 , 3 , 5 , 7 } .
Calculate the percentage of n ( A ) with respect to n ( U ) : 10 3 × 100 = 30% .
Explanation
Problem Analysis We are given the universal set U = { 1 , 2 , … , 10 } , the set A = { x : x ≤ 5 , x ∈ U , x is odd } and the set C = { z : z is a prime number , z ∈ U } . We need to find the cardinality of set A, the number of elements in A ∩ B ∩ C , show the relation of the sets U , A , B and C in a Venn diagram, and find the percentage of n ( A ) .
Elements of Set A First, let's list the elements of set A. Since A contains odd numbers less than or equal to 5 from the universal set U, we have A = { 1 , 3 , 5 } .
Cardinality of Set A The cardinality of set A, denoted as n ( A ) , is the number of elements in A. Therefore, n ( A ) = 3 .
Elements of Set C Next, let's list the prime numbers in the universal set U to define set C. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Thus, C = { 2 , 3 , 5 , 7 } .
Intersection of A, B, and C The problem statement is missing the definition of set B. Without knowing set B, we cannot determine A ∩ B ∩ C . We can only express it in terms of B. We know that A = { 1 , 3 , 5 } and C = { 2 , 3 , 5 , 7 } , so A ∩ C = { 3 , 5 } . Therefore, A ∩ B ∩ C = B ∩ { 3 , 5 } . The number of elements in A ∩ B ∩ C depends on the elements in set B.
Number of Elements in Intersection Since we don't know what set B is, we cannot find the number of elements in A ∩ B ∩ C .
Venn Diagram Representation To show the relation of the sets in a Venn diagram, we need to know set B. We can draw a Venn diagram with U as the universal set, and A and C as subsets. Set B would also be a subset, but its placement depends on its elements. Without knowing set B, we can only represent sets U, A, and C.
Percentage Calculation To find the percentage related to n ( A ) , we need to clarify what it is a percentage of. Assuming it is the percentage of the universal set U, we calculate n ( U ) n ( A ) × 100 . Since n ( A ) = 3 and n ( U ) = 10 , the percentage is 10 3 × 100 = 30% .
Percentage with Respect to U If the percentage of n ( A ) is with respect to another set, say set B, then we would need to know the cardinality of set B, n ( B ) , and the percentage would be n ( B ) n ( A ) × 100 = n ( B ) 3 × 100 . Without knowing set B, we cannot calculate this percentage. Assuming we want to find the percentage of n ( A ) with respect to n ( U ) , the percentage is 30% .
Final Answer Therefore, the cardinality of set A is 3. The number of elements in A ∩ B ∩ C cannot be determined without knowing set B. A Venn diagram can be drawn showing the relationship between U, A, and C, but the placement of set B depends on its elements. The percentage of n ( A ) with respect to n ( U ) is 30% .
Examples
Understanding sets and their relationships is crucial in many real-world scenarios. For example, in market research, we can define the universal set as all potential customers. Set A could be customers who prefer product A, set B could be customers who prefer product B, and set C could be customers who are price-sensitive. Analyzing the intersections of these sets helps companies tailor their marketing strategies. For instance, A ∩ B represents customers who like both products, and A ∩ B ∩ C represents price-sensitive customers who like both products. Knowing these segments allows for targeted promotions and product development.
