1. **Problem statement:** We are given exponential growth and decay rates for human and black bear populations in Florida from 1953 to 1993. (a) Find the human population in 1953 given it grew 8% per year to reach 13 million in 1993. (b) Find the black bear population in 1993 given it decreased 6% per year from 11,000 in 1953. (c) Find the year when the black bear population would fall below 110 if the trend continued. --- 2. **Formulas and rules:** Exponential growth/decay is modeled by: $$ P(t) = P_0 (1 + r)^t $$ where $P_0$ is the initial population, $r$ is the growth rate (positive for growth, negative for decay), and $t$ is the number of years. For decay, $r$ is negative, so $1 + r < 1$. --- 3. **Part (a) - Human population in 1953:** Given: - $P(40) = 13,000,000$ (population in 1993, 40 years after 1953) - Growth rate $r = 0.08$ Use formula: $$ 13,000,000 = P_0 (1 + 0.08)^{40} = P_0 (1.08)^{40} $$ Calculate $(1.08)^{40}$: $$ (1.08)^{40} \approx 21.7245 $$ Solve for $P_0$: $$ P_0 = \frac{13,000,000}{21.7245} $$ Intermediate step with cancellation: $$ P_0 = \frac{13,000,000}{\cancel{21.7245}} $$ Calculate: $$ P_0 \approx 598,500 $$ Rounded to nearest whole person: $$ \boxed{598500} $$ --- 4. **Part (b) - Black bear population in 1993:** Given: - Initial population $P_0 = 11,000$ - Decay rate $r = -0.06$ - Time $t = 40$ Use formula: $$ P(40) = 11,000 (1 - 0.06)^{40} = 11,000 (0.94)^{40} $$ Calculate $(0.94)^{40}$: $$ (0.94)^{40} \approx 0.0972 $$ Calculate population: $$ P(40) = 11,000 \times 0.0972 = 1069.2 $$ Rounded to nearest whole bear: $$ \boxed{1069} $$ --- 5. **Part (c) - Year when black bear population < 110:** We want $t$ such that: $$ 11,000 (0.94)^t < 110 $$ Divide both sides by 11,000: $$ (0.94)^t < \frac{110}{11,000} = 0.01 $$ Take natural logarithm: $$ \ln((0.94)^t) < \ln(0.01) $$ $$ t \ln(0.94) < \ln(0.01) $$ Since $\ln(0.94) < 0$, dividing reverses inequality: $$ t > \frac{\ln(0.01)}{\ln(0.94)} $$ Calculate: $$ \ln(0.01) \approx -4.6052 $$ $$ \ln(0.94) \approx -0.0619 $$ Calculate $t$: $$ t > \frac{-4.6052}{-0.0619} \approx 74.4 $$ Since $t$ is years after 1953, the year is: $$ 1953 + 75 = 2028 $$ Rounded to nearest whole year: $$ \boxed{2028} $$