1. **State the problem:** We need to calculate the adjusted exponential smoothing forecast for August 2015 using the given truck crossings data from January to July 2015, with smoothing constants $\alpha=0.2$ and $\beta=0.6$, and an initial trend of 10,000 trucks. 2. **Formula and explanation:** The adjusted exponential smoothing forecast uses the Holt’s linear trend method with the following equations: - Level: $$L_t = \alpha Y_t + (1 - \alpha)(L_{t-1} + T_{t-1})$$ - Trend: $$T_t = \beta (L_t - L_{t-1}) + (1 - \beta) T_{t-1}$$ - Forecast: $$F_{t+m} = L_t + m T_t$$ Here, $Y_t$ is the actual value at time $t$, $L_t$ is the level at time $t$, $T_t$ is the trend at time $t$, and $m$ is the number of periods ahead to forecast. 3. **Initial values:** Given: - $L_1 = Y_1 - T_1 = 127510 - 10000 = 117510$ - $T_1 = 10000$ 4. **Calculate $L_t$ and $T_t$ for each month from February to July:** For February ($t=2$): $$L_2 = 0.2 \times 115366 + 0.8 \times (117510 + 10000) = 23073.2 + 102008 = 125081.2$$ $$T_2 = 0.6 \times (125081.2 - 117510) + 0.4 \times 10000 = 0.6 \times 7571.2 + 4000 = 4542.72 + 4000 = 8542.72$$ For March ($t=3$): $$L_3 = 0.2 \times 128815 + 0.8 \times (125081.2 + 8542.72) = 25763 + 106498.74 = 132261.74$$ $$T_3 = 0.6 \times (132261.74 - 125081.2) + 0.4 \times 8542.72 = 0.6 \times 7180.54 + 3417.09 = 4308.32 + 3417.09 = 7725.41$$ For April ($t=4$): $$L_4 = 0.2 \times 123664 + 0.8 \times (132261.74 + 7725.41) = 24732.8 + 111189.7 = 135922.5$$ $$T_4 = 0.6 \times (135922.5 - 132261.74) + 0.4 \times 7725.41 = 0.6 \times 3660.76 + 3090.16 = 2196.46 + 3090.16 = 5286.62$$ For May ($t=5$): $$L_5 = 0.2 \times 124682 + 0.8 \times (135922.5 + 5286.62) = 24936.4 + 115375.3 = 140311.7$$ $$T_5 = 0.6 \times (140311.7 - 135922.5) + 0.4 \times 5286.62 = 0.6 \times 4389.2 + 2114.65 = 2633.52 + 2114.65 = 4748.17$$ For June ($t=6$): $$L_6 = 0.2 \times 135952 + 0.8 \times (140311.7 + 4748.17) = 27190.4 + 116831.1 = 144021.5$$ $$T_6 = 0.6 \times (144021.5 - 140311.7) + 0.4 \times 4748.17 = 0.6 \times 3709.8 + 1899.27 = 2225.88 + 1899.27 = 4125.15$$ For July ($t=7$): $$L_7 = 0.2 \times 123810 + 0.8 \times (144021.5 + 4125.15) = 24762 + 118498.1 = 143260.1$$ $$T_7 = 0.6 \times (143260.1 - 144021.5) + 0.4 \times 4125.15 = 0.6 \times (-761.4) + 1650.06 = -456.84 + 1650.06 = 1193.22$$ 5. **Calculate forecast for August 2015 ($m=1$):** $$F_{8} = L_7 + 1 \times T_7 = 143260.1 + 1193.22 = 144453.32$$ 6. **Final answer:** The adjusted exponential smoothing forecast for August 2015 is **144453.32** trucks (rounded to two decimal places).