Let's solve each part step by step: --- ### **a. Write the exponential function** Given: - Initial population in 2000, \( P_0 = 530,000 \) - Growth rate, \( r = 4.8\% = 0.048 \) - \( t \) = years since 2000 The formula for exponential growth is: \[ P(t) = P_0 e^{rt} \] Plug in the values: \[ P(t) = 530,000 \cdot e^{0.048t} \] --- ### **b. Find the rate at which the population is increasing in \( t \) years** This is the derivative of \( P(t) \): \[ P'(t) = \frac{d}{dt} \left( 530,000 \cdot e^{0.048t} \right) \] The derivative of \( ae^{kt} \) is \( ake^{kt} \): \[ P'(t) = 530,000 \cdot 0.048 \cdot e^{0.048t} \] \[ P'(t) = 25,440 \cdot e^{0.048t} \] --- ### **c. Find \( P'(21) \) (the rate in the year 2021, since \( t = 2021 - 2000 = 21 \))** Plug in \( t = 21 \): \[ P'(21) = 25,440 \cdot e^{0.048 \times 21} \] Calculate \( 0.048 \times 21 = 1.008 \): \[ P'(21) = 25,440 \cdot e^{1.008} \] Now calculate \( e^{1.008} \approx 2.740 \): \[ P'(21) \approx 25,440 \cdot 2.740 \approx 69,726 \] **Rounded to the nearest person:** \[ P'(21) \approx 69,726 \text{ people per year} \] --- ## **Final Answers:** **a.** \[ P(t) = 530,000 \cdot e^{0.048t} \] **b.** \[ P'(t) = 25,440 \cdot e^{0.048t} \text{ people per year} \] **c.** \[ P'(21) \approx 69,726 \text{ people per year} \] Let me know if you need any clarifications!

Let's go through each part with detailed calculations and explanations. --- ### **a. Write the exponential function \( P(t) \)** **Given:** - Population in 2000, \( P_0 = 530,000 \) - Annual growth rate, \( r = 4.8\% = 0.048 \) - \( t \) = number of years since 2000 **Standard exponential growth model:** \[ P(t) = P_0 \cdot e^{rt} \] **Substitute the known values:** \[ P(t) = 530,000 \cdot e^{0.048t} \] **Final answer for part (a):** \[ \boxed{ \textbf{a.} \quad P(t) = 530,000 \cdot e^{0.048t} } \] --- ### **b. Find the rate at which the population is increasing in \( t \) years** **Step 1:** Derive \( P(t) \) with respect to \( t \): \[ P(t) = 530,000 \cdot e^{0.048t} \] **Step 2:** Use the chain rule for differentiation: \[ P'(t) = 530,000 \cdot \frac{d}{dt} \left( e^{0.048t} \right) \] The derivative of \( e^{kt} \) is \( k e^{kt} \), so: \[ P'(t) = 530,000 \cdot 0.048 \cdot e^{0.048t} \] **Step 3:** Calculate the coefficient: \[ 530,000 \times 0.048 = 530,000 \times \frac{48}{1000} = \frac{530,000 \times 48}{1000} \] Calculate numerator: \[ 530,000 \times 48 = (530,000 \times 50) - (530,000 \times 2) = 26,500,000 - 1,060,000 = 25,440,000 \] Now divide by 1000: \[ \frac{25,440,000}{1000} = 25,440 \] **Thus:** \[ P'(t) = 25,440 \cdot e^{0.048t} \] **Final answer for part (b):** \[ \boxed{ \textbf{b.} \quad P'(t) = 25,440 \cdot e^{0.048t} \text{ (people per year)} } \] --- ### **c. Determine the rate of population increase in 2021** Since the year 2000 corresponds to \( t = 0 \), the year 2021 corresponds to: \[ t = 2021 - 2000 = 21 \] **Step 1:** Plug \( t = 21 \) into the derivative: \[ P'(21) = 25,440 \cdot e^{0.048 \times 21} \] Calculate the exponent: \[ 0.048 \times 21 = 1.008 \] **Step 2:** Calculate \( e^{1.008} \): Using a calculator or approximation: \[ e^{1.008} \approx e^{1} \times e^{0.008} \] We know: \[ e^{1} \approx 2.71828 \] \[ e^{0.008} \approx 1 + 0.008 + \frac{0.008^2}{2} = 1 + 0.008 + 0.000032 = 1.008032 \] Multiplying: \[ 2.71828 \times 1.008032 \approx 2.71828 \times (1 + 0.008032) \approx 2.71828 + 2.71828 \times 0.008032 \] Calculate: \[ 2.71828 \times 0.008032 \approx 0.0218 \] Adding: \[ 2.71828 + 0.0218 \approx 2.7401 \] Thus, \[ e^{1.008} \approx 2.740 \] **Step 3:** Calculate the rate: \[ P'(21) \approx 25,440 \times 2.740 = \] Multiply: \[ 25,440 \times 2.740 \] Break down: \[ 25,440 \times 2 = 50,880 \] \[ 25,440 \times 0.740 \approx 25,440 \times (0.7 + 0.04) = 25,440 \times 0.7 + 25,440 \times 0.04 \] Calculate: \[ 25,440 \times 0.7 = 17,808 \] \[ 25,440 \times 0.04 = 1,017.6 \] Add: \[ 17,808 + 1,017.6 = 18,825.6 \] Finally, sum all parts: \[ 50,880 + 18,825.6 = 69,705.6 \] **Round to the nearest person:** \[ \boxed{ \textbf{c.} \quad P'(21) \approx \boxed{69,706} \text{ people per year} } \] --- ## **Summary:** - **a.** \( P(t) = 530,000 \cdot e^{0.048t} \) - **b.** \( P'(t) = 25,440 \cdot e^{0.048t} \) - **c.** In 2021, the population increases at approximately **69,706 people per year**. Let me know if you'd like further clarification!