The average rate of change over 1 ≤ x ≤ 5 is -8.56 and the average rate of change over 6 ≤ x ≤ 8 is -18.1936.
Step-by-step explanation:
The average rate of change over 1 ≤ x ≤ 5 can be calculated using the formula: Average rate of change = (final value - initial value) / (final time - initial time). In this case, the initial value is 48. To find the final value, we can use the decay factor of 0.6. The final value after x time periods is 48 * (0.6)^x. Plugging in the values, the average rate of change over 1 ≤ x ≤ 5 is (48 * (0.6)^5 - 48) / (5 - 1) = (48 * 0.07776 - 48) / 4 ≈ -8.56.
The average rate of change over 6 ≤ x ≤ 8 can be calculated in the same way. In this case, the initial value is 48 * (0.6)^5 = 18.8544. To find the final value, we can use the decay factor of 0.6. The final value after x time periods is 18.8544 * (0.6)^(x-5). Plugging in the values, the average rate of change over 6 ≤ x ≤ 8 is (18.8544 * (0.6)^8 - 18.8544) / (8 - 6) = (18.8544 * 0.046656 - 18.8544) / 2 ≈ -18.1936.