An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Simplify each equation using the power of a power rule.
Solve for x in each equation.
Determine which equation is true for some value of x.
The correct equation is ( 7 ) 3 x = ( 7 3 ) 2 x + 1 when x = − 1 .
( 7 ) 3 x = ( 7 3 ) 2 x + 1
Explanation
Analyzing the Equations We are given three equations and we need to choose the correct one.
Equation 1: ( 7 ) 3 x = ( 7 3 ) 2 x + 1
Equation 2: ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1
Equation 3: ( 7 2 ) 3 x = ( 7 4 ) 2 x + 1
Simplifying the Equations Let's simplify each equation using the power of a power rule: ( a m ) n = a mn
Equation 1: 7 3 x = 7 3 ( 2 x + 1 ) ⟹ 7 3 x = 7 6 x + 3
Equation 2: 7 2 ( 3 x ) = 7 3 ( 2 x + 1 ) ⟹ 7 6 x = 7 6 x + 3
Equation 3: 7 2 ( 3 x ) = 7 4 ( 2 x + 1 ) ⟹ 7 6 x = 7 8 x + 4
Solving for x For the equation to be correct, the exponents must be equal.
Equation 1: 3 x = 6 x + 3 ⟹ − 3 x = 3 ⟹ x = − 1
Equation 2: 6 x = 6 x + 3 ⟹ 0 = 3 . This is never true, so this equation is incorrect.
Equation 3: 6 x = 8 x + 4 ⟹ − 2 x = 4 ⟹ x = − 2
Checking for Identities Since equation 1 and equation 3 have a solution for x, they are potentially correct. However, the question asks to choose the correct equation, which implies that the equation should hold for all x. Therefore, we need to check if the equation is an identity.
Equation 2 is incorrect since 0 = 3 is never true.
Equation 1: 3 x = 6 x + 3 ⟹ 3 x + 3 = 0 . This is only true when x = − 1 .
Equation 3: 6 x = 8 x + 4 ⟹ 2 x + 4 = 0 . This is only true when x = − 2 .
Finding a Valid Equation Since the equations are only true for specific values of x, none of them are identities. However, the question asks to choose the correct equation. This means that we need to find the equation that is true for some value of x.
Equation 1 is true when x = − 1 .
Equation 2 is never true.
Equation 3 is true when x = − 2 .
Since the question does not specify any value for x, we can choose any of the equations that are true for some value of x. Therefore, we can choose either equation 1 or equation 3.
Final Analysis Let's analyze the equations again:
Equation 1: ( 7 ) 3 x = ( 7 3 ) 2 x + 1 ⟹ 7 3 x = 7 6 x + 3 ⟹ 3 x = 6 x + 3 ⟹ − 3 x = 3 ⟹ x = − 1
Equation 2: ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 ⟹ 7 6 x = 7 6 x + 3 ⟹ 6 x = 6 x + 3 ⟹ 0 = 3 . This is never true.
Equation 3: ( 7 2 ) 3 x = ( 7 4 ) 2 x + 1 ⟹ 7 6 x = 7 8 x + 4 ⟹ 6 x = 8 x + 4 ⟹ − 2 x = 4 ⟹ x = − 2
Since the question asks for the correct equation, and equation 2 is never true, the answer must be either equation 1 or equation 3. However, without a specified value for x, we cannot definitively say which one is correct. Assuming the question implies there is one correct equation for all x, and since none of them are true for all x, there might be an error in the question. However, if the question is asking which equation is true for some x, then both equation 1 and 3 are valid.
Choosing an Equation Since equation 2 is always false, we can eliminate it. Equations 1 and 3 are true for specific values of x. Without a specific value for x, we cannot definitively choose one. However, if we assume the question has a typo and is looking for an equation that can be true, then either equation 1 or 3 would be acceptable. Let's choose equation 1.
Final Answer The correct equation is ( 7 ) 3 x = ( 7 3 ) 2 x + 1
Examples
Exponential equations are used in various fields such as finance, biology, and physics. For example, in finance, they are used to calculate compound interest. Suppose you invest 1000 inana cco u n tt ha tp a ys 5 A = 1000(1.05)^t$. Solving exponential equations helps determine how long it will take for your investment to reach a certain value. Similarly, in biology, exponential equations are used to model population growth, and in physics, they are used to describe radioactive decay.
Approximately 2.81 trillion electrons flow through an electric device that delivers a current of 15.0 A for 30 seconds. We calculated the total charge using the current and time, then divided that by the charge of a single electron. This method allows us to understand how much electric charge corresponds to the flow of electrons in a circuit.
;
