1. The problem states that the salvage value $S(t)$ of the company jet after $t$ years is given by: $$S(t) = 700,000 \times (0.8)^t$$ 2. We are asked to find the rate of depreciation in dollars per year after 1 year and after 5 years. The rate of depreciation is the negative of the derivative of $S(t)$ with respect to $t$ because the salvage value decreases over time. 3. Differentiate $S(t)$: $$\frac{dS}{dt} = 700,000 \times \frac{d}{dt} \left((0.8)^t\right)$$ 4. Recall that $\frac{d}{dt} a^t = a^t \ln(a)$, so: $$\frac{dS}{dt} = 700,000 \times (0.8)^t \times \ln(0.8)$$ 5. Calculate the rate of depreciation after 1 year ($t=1$): $$\frac{dS}{dt}\Big|_{t=1} = 700,000 \times (0.8)^1 \times \ln(0.8)$$ Compute $(0.8)^1 = 0.8$ and $\ln(0.8) \approx -0.2231435513$: $$700,000 \times 0.8 \times (-0.2231435513) = -124,960.39$$ Note the negative sign means depreciation, so the rate is $-124,960.39$ dollars per year after 1 year. 6. Calculate the rate of depreciation after 5 years ($t=5$): $$\frac{dS}{dt}\Big|_{t=5} = 700,000 \times (0.8)^5 \times \ln(0.8)$$ Compute $(0.8)^5 = 0.32768$ and $\ln(0.8) \approx -0.2231435513$: $$700,000 \times 0.32768 \times (-0.2231435513) = -51,132.96$$ 7. Therefore, the rate of depreciation after 5 years is approximately $-51,132.96$ dollars per year. **Final answers:** - After 1 year: $-124,960.39$ dollars per year - After 5 years: $-51,132.96$ dollars per year