1. **State the problem:** We have an exponential decay function with initial value 500 and decay rate 15%. We want to compare the average rate of change on intervals $0 < x < 4$ and $4 < x < 8$, and predict the behavior beyond $x=8$. 2. **Write the function:** The general form for exponential decay is $$f(x) = a(1 - r)^x$$ where $a$ is the initial value and $r$ is the decay rate. Here, $$a = 500, \quad r = 0.15,$$ so $$f(x) = 500(0.85)^x.$$ 3. **Calculate function values at key points:** $$f(0) = 500(0.85)^0 = 500,$$ $$f(4) = 500(0.85)^4 = 500 \times 0.52200625 = 261.003,$$ $$f(8) = 500(0.85)^8 = 500 \times 0.27249052 = 136.245.$$ 4. **Calculate average rate of change on $0 < x < 4$:** $$\text{Average rate} = \frac{f(4) - f(0)}{4 - 0} = \frac{261.003 - 500}{4} = \frac{-238.997}{4} = -59.74925 \approx -60.$$ 5. **Calculate average rate of change on $4 < x < 8$:** $$\text{Average rate} = \frac{f(8) - f(4)}{8 - 4} = \frac{136.245 - 261.003}{4} = \frac{-124.758}{4} = -31.1895 \approx -31.$$ 6. **Interpretation for intervals beyond $x=8$:** Since the function is exponential decay, the rate of change remains negative but its magnitude decreases as $x$ increases. This means the function decreases more slowly over time, approaching zero but never becoming negative. **Final answer:** - Average rate of change on $0 < x < 4$ is approximately $-60$. - Average rate of change on $4 < x < 8$ is approximately $-31$. - Beyond $x=8$, the rate of change stays negative but lies between 0 and $-15$, decreasing in magnitude as $x$ increases.