An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Identify the exponential function as f ( x ) = a ⋅ b x , where b is the decay factor.
Determine the initial value a from the table: f ( 0 ) = a = 6 .
Use another point from the table, such as (1, 2), to solve for b : 6 ⋅ b = 2 , so b = 3 1 .
The decay factor is 3 1 .
Explanation
Understanding the Problem We are given a table of values for an exponential function and asked to find the decay factor. The table provides the following data points: (-1, 18), (0, 6), (1, 2), and (2, 2/3).
Identifying the Exponential Function An exponential function can be written in the form f ( x ) = a ⋅ b x , where a is the initial value and b is the decay/growth factor. In this case, since the values of f ( x ) are decreasing as x increases, we expect b to be a decay factor (i.e., 0 < b < 1 ).
Finding the Initial Value From the table, we can see that f ( 0 ) = 6 . Plugging this into our equation, we get f ( 0 ) = a ⋅ b 0 = a ⋅ 1 = a . Therefore, a = 6 , and our function is f ( x ) = 6 ⋅ b x .
Solving for the Decay Factor Now we can use another point from the table to solve for b . Let's use the point (1, 2). We have f ( 1 ) = 2 , so 6 ⋅ b 1 = 2 . Dividing both sides by 6, we get b = 6 2 = 3 1 .
Finding the Common Ratio Alternatively, we can find the ratio between consecutive y-values. f ( − 1 ) f ( 0 ) = 18 6 = 3 1 , f ( 0 ) f ( 1 ) = 6 2 = 3 1 , f ( 1 ) f ( 2 ) = 2 2/3 = 3 1 . The common ratio is 3 1 , which is the decay factor.
Verification Thus, the decay factor of the exponential function is 3 1 . We can verify this by plugging the decay factor back into the equation and checking if it matches the given data points. f ( x ) = 6 ⋅ ( 3 1 ) x . f ( − 1 ) = 6 ⋅ ( 3 1 ) − 1 = 6 ⋅ 3 = 18 . f ( 0 ) = 6 ⋅ ( 3 1 ) 0 = 6 . f ( 1 ) = 6 ⋅ ( 3 1 ) 1 = 2 . f ( 2 ) = 6 ⋅ ( 3 1 ) 2 = 6 ⋅ 9 1 = 3 2 .
Final Answer The decay factor of the exponential function represented by the table is 3 1 .
Examples
Exponential decay is a concept that appears in various real-world scenarios. For instance, the depreciation of a car's value over time can be modeled using an exponential decay function. If a car is bought for 20 , 000 an d i t s v a l u e d ecre a ses b y 15 V(t) = 20000 \cdot (0.85)^t , w h ere t$ is the number of years since the purchase. Similarly, the cooling of an object, the decay of radioactive substances, and the decrease in the intensity of light as it passes through a medium can all be modeled using exponential decay functions. Understanding exponential decay helps in predicting and analyzing these phenomena.
A current of 15.0 A for 30 seconds results in a total charge of 450 C , which is equivalent to approximately 2.81 × 1 0 21 electrons flowing through the device.
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