1. **Problem Statement:** We are given two exponential decay functions: $$P(t) = 20000(0.92)^t$$ and $$Q(t) = 20000((0.9)(0.92))^t$$ We want to understand which function correctly models the population decay and why. 2. **Understanding the Functions:** Both functions represent exponential decay because their bases (0.92 and 0.9 \times 0.92) are less than 1. 3. **Simplify Q(t):** $$Q(t) = 20000(0.9 \times 0.92)^t = 20000(0.828)^t$$ 4. **Compare the decay rates:** - In $P(t)$, the decay factor per year is $0.92$. - In $Q(t)$, the decay factor per year is $0.828$. Since $0.828 < 0.92$, $Q(t)$ decays faster than $P(t)$. 5. **Which is correct?** - If the population decreases by 8% each year, $P(t)$ with $0.92^t$ is correct. - If the population decreases by 17.2% each year (since $1 - 0.828 = 0.172$), then $Q(t)$ is correct. 6. **Why multiply 0.9 and 0.92?** - Multiplying the two decay factors assumes two independent decay processes happening simultaneously each year: one with factor 0.9 and another with 0.92. - This results in a combined decay factor of $0.9 \times 0.92 = 0.828$. 7. **Final answer:** - The right answer depends on the context. - If the problem states a single decay rate of 8%, use $P(t) = 20000(0.92)^t$. - If the problem involves two sequential decay processes of 10% and 8%, use $Q(t) = 20000(0.828)^t$. **Summary:** $$Q(t) = 20000(0.828)^t$$ decays faster than $$P(t) = 20000(0.92)^t$$ because it combines two decay factors. Choose the model based on the actual decay scenario.