An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
The function y = ( 3 1 ) x is decreasing because its base is between 0 and 1.
The function has no x -intercept.
The y -intercept is found by setting x = 0 , giving y = ( 3 1 ) 0 = 1 , so the y -intercept is ( 0 , 1 ) .
The range of the function is all positive real numbers, ( 0 , ∞ ) .
The correct statements are that the function is decreasing and the y -intercept is ( 0 , 1 ) .
Explanation
Analyzing the Problem We are given the function y = ( 3 1 ) x and asked to determine which statements about its graph are true. Let's analyze each statement.
Increasing or Decreasing To determine if the function is increasing or decreasing, we can analyze its base. Since the base 3 1 is between 0 and 1, the function is a decreasing exponential function. As x increases, y decreases.
Finding the x-intercept To find the x -intercept, we set y = 0 and solve for x . However, the function y = ( 3 1 ) x never equals 0 for any real value of x . Therefore, there is no x -intercept.
Finding the y-intercept To find the y -intercept, we set x = 0 and solve for y . y = ( 3 1 ) 0 = 1 So, the y -intercept is ( 0 , 1 ) .
Determining the Range The range of the function y = ( 3 1 ) x consists of all possible y values. Since the base is positive, the function is always positive. As x approaches infinity, y approaches 0, but never reaches it. As x approaches negative infinity, y approaches infinity. Therefore, the range of the function is all positive real numbers, or ( 0 , ∞ ) .
Conclusion Based on our analysis, the correct statements are:
The function is decreasing.
The y -intercept is ( 0 , 1 ) .
Examples
Exponential functions like y = ( 3 1 ) x are used to model various real-world phenomena, such as radioactive decay. Imagine you have a sample of a radioactive substance that decays at a rate of 3 2 per unit of time. The amount of the substance remaining after x units of time can be modeled by the function y = A 0 ( 3 1 ) x , where A 0 is the initial amount of the substance. Understanding the properties of exponential functions helps scientists predict how much of the substance will remain after a certain period.
To calculate the number of electrons flowing through a device with a current of 15.0 A over 30 seconds, we find the total charge is 450 C . Dividing this by the charge of one electron, we discover approximately 2.81 × 1 0 21 electrons flow through the device.
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