Which of the following represents the concept of exponential decay?

Which of the following represents the concept of exponential decay?

• A. P = A(1+r)^t
• B. y = a(1-r)^t
• C. y = A(b)^x
• D. y = A(1+r)^x

Answer: (B)

The concept of exponential decay is characterized by a quantity decreasing at a rate that is proportional to its current value over time. The correct choice, represented by \( y = a(1 - r)^t \), clearly reflects this behavior. In this equation, \( a \) signifies the initial amount, \( r \) is a positive rate of decay (indicating that the quantity is decreasing), and \( t \) represents time. As \( t \) increases, the term \( (1 - r) \) is raised to a higher power. Since \( 1 - r \) is less than 1 (because \( r \) is positive), the overall value of \( y \) diminishes, illustrating the nature of exponential decay. This structure is distinct from the other options. For instance, options presenting the form \( A(1 + r)^t \) or \( A(1 + r)^x \) would represent growth rather than decay, as they describe scenarios where the quantity increases over time. Meanwhile, the expression \( y = A(b)^x \) can denote either growth or decay depending on the value of \( b \). If \( b \) is less than 1, it resembles decay,

The concept of exponential decay is characterized by a quantity decreasing at a rate that is proportional to its current value over time. The correct choice, represented by ( y = a(1 - r)^t ), clearly reflects this behavior.

In this equation, ( a ) signifies the initial amount, ( r ) is a positive rate of decay (indicating that the quantity is decreasing), and ( t ) represents time. As ( t ) increases, the term ( (1 - r) ) is raised to a higher power. Since ( 1 - r ) is less than 1 (because ( r ) is positive), the overall value of ( y ) diminishes, illustrating the nature of exponential decay.

This structure is distinct from the other options. For instance, options presenting the form ( A(1 + r)^t ) or ( A(1 + r)^x ) would represent growth rather than decay, as they describe scenarios where the quantity increases over time. Meanwhile, the expression ( y = A(b)^x ) can denote either growth or decay depending on the value of ( b ). If ( b ) is less than 1, it resembles decay,