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Question Number 131481 by bemath last updated on 05/Feb/21

$$\mathrm{In}1790\mathrm{the}\mathrm{population}\mathrm{of}\mathrm{the}$$$$\mathrm{United}\mathrm{States}\mathrm{was}3.93\mathrm{million}$$$$\mathrm{and}\mathrm{in}1890\mathrm{it}\mathrm{was}62.98\mathrm{million}$$$$\mathrm{Using}\mathrm{the}\mathrm{Malthusian}\mathrm{model},$$$$\mathrm{estimate}\mathrm{the}\mathrm{U}.\mathrm{S}\mathrm{population}\mathrm{as}$$$$\mathrm{a}\mathrm{function}\mathrm{of}\mathrm{time}$$

Answered by EDWIN88 last updated on 05/Feb/21

$$\mathrm{we}\mathrm{set}\mathrm{t}=0\mathrm{to}\mathrm{be}\mathrm{year}1790\mathrm{then}\mathrm{we}\mathrm{have}$$$$\mathrm{p}\left(\mathrm{t}\right)=3.93\times {\mathrm{e}}^{\mathrm{kt}};\mathrm{where}\mathrm{p}\left(\mathrm{t}\right)\mathrm{is}\mathrm{population}$$$$\mathrm{in}\mathrm{millions}.\mathrm{and}\mathrm{p}\left(100\right)=62.98=3.93\times {\mathrm{e}}^{100\mathrm{k}}$$$$\mathrm{we}\mathrm{find}\mathrm{k}=\frac{\ln (62.98)-\ln (3.98)}{100}\approx 0.027742$$$$\mathrm{so}\mathrm{p}\left(\mathrm{t}\right)=3.93\times {\mathrm{e}}^{0.027742\mathrm{t}}$$

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