Which of the following represents the concept of exponential 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,