An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate the ratio between consecutive y values: 10 1 2 1 = 5 , 2 1 2 5 = 5 , 2 5 2 25 = 5 , 2 25 2 125 = 5 .
Since the ratios are constant, the function is exponential.
The constant ratio represents the rate of change.
The rate of change of the function is 5 .
Explanation
Analyzing the Data We are given a table of x and y values and asked to find the rate of change of the function described by the table. The x values are -1, 0, 1, 2, 3 and the corresponding y values are 10 1 , 2 1 , 2 5 , 2 25 , 2 125 .
Finding the Ratio To find the rate of change, we need to determine the relationship between consecutive y values. We can do this by calculating the ratio between consecutive y values.
Calculating the Rate of Change Let's calculate the ratios:
10 1 2 1 = 2 1 × 1 10 = 5
2 1 2 5 = 2 5 × 1 2 = 5
2 5 2 25 = 2 25 × 5 2 = 5
2 25 2 125 = 2 125 × 25 2 = 5
Since the ratio between consecutive y values is constant and equal to 5, the function is exponential, and the rate of change is 5.
Final Answer The rate of change of the function is 5.
Examples
Exponential functions are used to model population growth, radioactive decay, and compound interest. For example, if a population of bacteria doubles every hour, the population can be modeled by an exponential function with a base of 2. Similarly, if a radioactive substance decays at a rate of 10% per year, the amount of the substance remaining can be modeled by an exponential function with a base of 0.9.
