1. **State the problem:** We have an exponential decay function with initial value 500 and decay rate 15%. We want to find the average rate of change on intervals $0 < x < 4$ and $4 < x < 8$, then predict the behavior beyond $x=8$. 2. **Write the function:** The general form for exponential decay is $$y = y_0 (1 - r)^x$$ where $y_0$ is the initial value and $r$ is the decay rate. Here, $$y = 500 (1 - 0.15)^x = 500 (0.85)^x$$ 3. **Calculate values at interval endpoints:** - At $x=0$: $$y(0) = 500 (0.85)^0 = 500$$ - At $x=4$: $$y(4) = 500 (0.85)^4 = 500 \times 0.85^4$$ Calculate $0.85^4$: $$0.85^4 = 0.85 \times 0.85 \times 0.85 \times 0.85 = 0.52200625$$ So, $$y(4) = 500 \times 0.52200625 = 261.003125$$ - At $x=8$: $$y(8) = 500 (0.85)^8 = 500 \times (0.85^4)^2 = 500 \times 0.52200625^2$$ Calculate $0.52200625^2$: $$0.52200625^2 = 0.272489$$ So, $$y(8) = 500 \times 0.272489 = 136.2445$$ 4. **Calculate average rate of change for $0 < x < 4$:** $$\text{Average rate} = \frac{y(4) - y(0)}{4 - 0} = \frac{261.003125 - 500}{4} = \frac{-238.996875}{4} = -59.7492$$ Rounded to nearest integer: $$-60$$ 5. **Calculate average rate of change for $4 < x < 8$:** $$\text{Average rate} = \frac{y(8) - y(4)}{8 - 4} = \frac{136.2445 - 261.003125}{4} = \frac{-124.758625}{4} = -31.1897$$ Rounded to nearest integer: $$-31$$ 6. **Interpretation:** The average rate of change is negative in both intervals, indicating decay. The magnitude of the rate of change decreases from about $-60$ to $-31$, meaning the function is decaying more slowly over time. 7. **Prediction beyond $x=8$:** Since the function is exponential decay, the rate of change will continue to decrease in magnitude (become less negative) as $x$ increases, approaching zero but never becoming positive. The function flattens out as it approaches zero. **Final answers:** - Average rate of change for $0 < x < 4$ is approximately $-60$ - Average rate of change for $4 < x < 8$ is approximately $-31$ - Beyond $x=8$, the rate of change will continue to approach zero from the negative side, meaning the decay slows down over time.