An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The initial value for 60 = 5 ( 1.07 ) x is 5, and it represents growth at a rate of 7%.
The initial value for 5 = 60 ( 0.93 ) x is 60, and it represents decay at a rate of 7%.
Therefore, the completed tables have the following values: Initial Value: 5 (Growth); Initial Value: 60 (Decay).
The growth/decay rates are both 7%, but one represents growth and the other represents decay. 7%
Explanation
Understanding Exponential Equations We are given two exponential equations and need to identify the initial value, growth/decay, and growth/decay rate for each. The general form of an exponential equation is y = a ( b ) x , where 'a' is the initial value, and 'b' determines growth or decay. If b > 1, it's growth; if 0 < b < 1, it's decay. The growth/decay rate is given by ∣ b − 1∣ .
Analyzing the First Equation For the equation 60 = 5 ( 1.07 ) x :
The initial value is the coefficient 'a', which is 5.
The base 'b' is 1.07, which is greater than 1, so it represents growth.
The growth rate is ( 1.07 − 1 ) = 0.07 , which is 7%.
Analyzing the Second Equation For the equation 5 = 60 ( 0.93 ) x :
The initial value is the coefficient 'a', which is 60.
The base 'b' is 0.93, which is between 0 and 1, so it represents decay.
The decay rate is ∣0.93 − 1∣ = ∣ − 0.07∣ = 0.07 , which is 7%.
Filling in the Tables Based on the analysis above, we can fill in the tables as follows:
Table 1:
Initial Value: 5
Growth or Decay: Growth
Growth/Decay Rate: 7%
Table 2:
Initial Value: 60
Growth or Decay: Decay
Growth/Decay Rate: 7%
Examples
Exponential equations are used in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. For example, if a population starts with 500 individuals and grows at a rate of 3% per year, the population after 'x' years can be modeled by the equation P = 500 ( 1.03 ) x . Similarly, if you invest 1000 inana cco u n tt ha t e a r n s 5 A = 1000(1.05)^x$. Understanding exponential equations helps in making predictions and informed decisions in these situations.
The device with a current of 15.0 A for 30 seconds delivers a total charge of 450 C . This amount of charge corresponds to approximately 2.81 × 1 0 21 electrons flowing through the device. The calculation utilized the relationship between current, charge, and the charge of a single electron.
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