1. Problem: Find the rate of depreciation after 1 year for the salvage value of a company jet given by the formula $$S(t) = 500000(0.9)^t$$ where $S(t)$ is the salvage value in dollars after $t$ years. 2. The rate of depreciation is the rate of change of the salvage value over time, i.e., the derivative $$\frac{dS}{dt}$$. 3. Differentiate $$S(t) = 500000(0.9)^t$$ with respect to $t$: $$\frac{dS}{dt} = 500000 \cdot \frac{d}{dt}(0.9)^t = 500000 (0.9)^t \ln(0.9)$$ 4. Evaluate the derivative at $t=1$: $$\frac{dS}{dt} \bigg|_{t=1} = 500000 (0.9)^1 \ln(0.9) = 500000 \cdot 0.9 \cdot \ln(0.9)$$ 5. Calculate numeric value: $$\ln(0.9) \approx -0.105360516$$ $$500000 \cdot 0.9 = 450000$$ $$450000 \times -0.105360516 \approx -47412.23$$ 6. Interpretation: The negative sign indicates depreciation. The rate of depreciation after 1 year is approximately $47412.23$ dollars per year. **Final answer:** The rate of depreciation after 1 year is about 47412.23 dollars per year.