1. **State the problem:** We know the Amazon rainforest decreased by 20% over 50 years, and its area after 50 years is 3,290,125 km^2. We want to find: a) The predicted area after 100 more years if the rate stays the same. b) The estimated area at the start of the 50-year period. 2. **Formula and explanation:** The decrease is exponential decay, so the area after time $t$ can be modeled as: $$ A(t) = A_0 \times (1 - r)^t $$ where $A_0$ is the initial area, $r$ is the decay rate per 50 years (20% = 0.20), and $t$ is the number of 50-year periods. 3. **Find the initial area $A_0$:** Given $A(1) = 3,290,125$ km^2 after 1 period (50 years), $$ 3,290,125 = A_0 \times (1 - 0.20)^1 = A_0 \times 0.80 $$ Divide both sides by 0.80: $$ A_0 = \frac{3,290,125}{0.80} $$ Show cancellation: $$ A_0 = \frac{3,290,125}{\cancel{0.80}} \times \frac{\cancel{1}}{1} = 4,112,656.25 $$ Rounded to nearest km^2: $$ A_0 \approx 4,112,656 \text{ km}^2 $$ 4. **Predict area after 100 more years (2 periods):** We want $A(3)$ because 100 years later means 2 more periods after the first 50 years, so total $t=3$. $$ A(3) = A_0 \times (0.80)^3 $$ Calculate: $$ A(3) = 4,112,656.25 \times 0.512 = 2,104,680.32 $$ Rounded: $$ A(3) \approx 2,104,680 \text{ km}^2 $$ **Final answers:** a) Predicted area after 100 more years: $2,104,680$ km$^2$ b) Estimated area at start: $4,112,656$ km$^2$