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Home » To minimize the distance, minimize $\|\overrightarrowCP\|^2$: - Get link 4share

To minimize the distance, minimize $\|\overrightarrowCP\|^2$: - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 A research project tracks invasive plant spread modeled by \( A(t) = 150e^{0.1t} \), where \( A \) is area in hectares. How long until the area triples from its initial value? Use \( \ln(3) \approx 1.0986 \).

📌 Initial area at \( t = 0 \): \( A(0) = 150 \)

📌 Tripled: \( 450 = 150e^{0.1t} \Rightarrow 3 = e^{0.1t} \Rightarrow \ln 3 = 0.1t \Rightarrow t = \frac{1.0986}{0.1} = 10.986 \approx 11 \) years

📌 In a carbon sequestration study, a forest absorbs CO₂ at a rate proportional to its current biomass. If biomass grows from 500 tons to 800 tons in 6 years, what is the approximate annual growth rate (as a percent), using the formula \( B(t) = B_0 e^{rt} \)?

📌 \( 800 = 500 e^{6r} \Rightarrow \frac{8}{5} = 1.6 = e^{6r} \Rightarrow \ln(1.6) = 6r \)

📌 \( \ln(1.6) = \ln(8/5) = \ln 8 - \ln 5 = 3\ln 2 - \ln(5) \approx 3(0.6931) - 1.6094 = 2.0793 - 1.6094 = 0.4699 \)

📌 So \( r = \frac{0.4699}{6} \approx 0.0783 \) → 7.83%

📌 Rounded to nearest percent: 8%

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