An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Test each function with x = 0 to eliminate options.
Test the remaining functions with x = 1 to further narrow down the options.
Verify the remaining function with x = 2 , x = − 1 , and x = − 2 .
The exponential function represented by the values in the table is f ( x ) = 4 ( 2 1 ) x .
Explanation
Understanding the Problem We are given a table of values for an exponential function f ( x ) and four possible functions. Our goal is to determine which of the given functions matches the values in the table.
Solution Plan We will test each of the given functions with the values from the table, starting with x = 0 since f ( 0 ) is easy to compute.
Testing x=0
For f ( x ) = 2 1 ( 4 ) x , f ( 0 ) = 2 1 ( 4 ) 0 = 2 1 ( 1 ) = 2 1 = 4 . So this function is not the correct one.
For f ( x ) = 4 ( 4 ) x , f ( 0 ) = 4 ( 4 ) 0 = 4 ( 1 ) = 4 . This is consistent with the table.
For f ( x ) = 4 ( 2 1 ) x , f ( 0 ) = 4 ( 2 1 ) 0 = 4 ( 1 ) = 4 . This is consistent with the table.
For f ( x ) = 2 1 ( 2 1 ) x , f ( 0 ) = 2 1 ( 2 1 ) 0 = 2 1 ( 1 ) = 2 1 = 4 . So this function is not the correct one.
Narrowing Down the Options Now we have two possible functions: f ( x ) = 4 ( 4 ) x and f ( x ) = 4 ( 2 1 ) x . Let's test x = 1 .
Testing x=1
For f ( x ) = 4 ( 4 ) x , f ( 1 ) = 4 ( 4 ) 1 = 16 = 2 . So this function is not the correct one.
For f ( x ) = 4 ( 2 1 ) x , f ( 1 ) = 4 ( 2 1 ) 1 = 4 ( 2 1 ) = 2 . This is consistent with the table.
Verification Since f ( x ) = 4 ( 2 1 ) x is the only remaining candidate, let's test a few more values to be sure.
More Verification
Test x = 2 . For f ( x ) = 4 ( 2 1 ) x , f ( 2 ) = 4 ( 2 1 ) 2 = 4 ( 4 1 ) = 1 . This is consistent with the table.
Test x = − 1 . For f ( x ) = 4 ( 2 1 ) x , f ( − 1 ) = 4 ( 2 1 ) − 1 = 4 ( 2 ) = 8 . This is consistent with the table.
Test x = − 2 . For f ( x ) = 4 ( 2 1 ) x , f ( − 2 ) = 4 ( 2 1 ) − 2 = 4 ( 4 ) = 16 . This is consistent with the table.
Final Answer Therefore, the correct function is f ( x ) = 4 ( 2 1 ) x .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that pays compound interest, the amount of money you have in the account will grow exponentially over time. Understanding exponential functions can help you make informed decisions about your investments and other financial matters. Also, exponential functions are used in physics to model the decay of radioactive substances. The half-life of a radioactive substance is the time it takes for half of the substance to decay. The amount of radioactive substance remaining after a certain amount of time can be calculated using an exponential function.
The total charge delivered by a device with a current of 15.0 A over 30 seconds is 450 C . This charge corresponds to approximately 2.81 × 1 0 21 electrons flowing through the device. The calculation involves using the relationship between current, charge, and the charge of an electron.
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