The rate at which the population of a certain city increases is proportional to the population. If there were 125,000 people in the city in 1985 and 140,000 in 2000 , what population can be predicted for the year 2025 ? Do not round the value of \( k \), but round your final answer to the nearest whole number.
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To solve for the predicted population in 2025, we can use the formula for exponential growth, which is \( P(t) = P_0 e^{kt} \). 1. First, we need to calculate the value of \( k \) using the information provided. We have: - \( P_0 = 125,000 \) (the population in 1985) - \( P(15) = 140,000 \) (the population in 2000, which is 15 years after 1985) Using the equation: \[ 140,000 = 125,000 e^{15k} \] Dividing both sides by 125,000: \[ \frac{140,000}{125,000} = e^{15k} \] \[ 1.12 = e^{15k} \] Taking the natural logarithm: \[ \ln(1.12) = 15k \] \[ k = \frac{\ln(1.12)}{15} \approx 0.007995 \] 2. Now, we can predict the population in 2025, which is 40 years after 1985: \[ P(40) = 125,000 e^{40k} \] Substituting \( k \): \[ P(40) = 125,000 e^{40 \times 0.007995} \] \[ P(40) = 125,000 e^{0.3198} \] \[ P(40) \approx 125,000 \times 1.3766 \approx 172,075 \] Rounding to the nearest whole number gives us a predicted population of approximately **172,075** for the year 2025. Happy predicting! 🌟 The world of exponential growth is as fascinating as it is useful—every new statistic can help shape the future!