Question

1. $y = 8(1 - 0.15)^t$

Explanation:

Step1: Simplify the base

First, calculate the value inside the parentheses: \(1 - 0.15 = 0.85\). So the function becomes \(y = 8(0.85)^t\). This is a general form of an exponential decay function, where the initial amount (when \(t = 0\)) is the coefficient of the exponential term, and the base \(0.85\) (which is less than 1) indicates decay. If we want to analyze the rate of decay, the decay rate \(r\) is related to the base \(b\) by \(b=1 - r\), so here \(r = 0.15\) or \(15\%\) decay per time unit \(t\). If we were to evaluate the function at a specific \(t\), say \(t = 1\), we would calculate \(y=8\times0.85 = 6.8\); at \(t = 2\), \(y = 8\times(0.85)^2=8\times0.7225 = 5.78\), and so on.

Answer:

The function \(y = 8(1 - 0.15)^t\) simplifies to \(y = 8(0.85)^t\), representing an exponential decay function with an initial value of \(8\) and a decay rate of \(15\%\) per unit time \(t\). If evaluating at a specific \(t\), substitute the value of \(t\) into \(y = 8(0.85)^t\) to find \(y\). For example, at \(t = 0\), \(y = 8\); at \(t = 1\), \(y = 6.8\); at \(t = 2\), \(y = 5.78\), etc.