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From 1790 to 1860 , the United States population, in thousands, is modeled by the equation 
P=4000(1.031)^(t) where 
t is the number of years since 1790 .
a. About how many people were living in the U.S. in 1790? What about in 1860 ? Show your reasoning.

From 17901790 to 18601860 , the United States population, in thousands, is modeled by the equation P=4000(1.031)t P=4000(1.031)^{t} where t t is the number of years since 17901790 .\newlinea. About how many people were living in the U.S. in 17901790? What about in 18601860 ? Show your reasoning.

Full solution

Q. From 17901790 to 18601860 , the United States population, in thousands, is modeled by the equation P=4000(1.031)t P=4000(1.031)^{t} where t t is the number of years since 17901790 .\newlinea. About how many people were living in the U.S. in 17901790? What about in 18601860 ? Show your reasoning.
  1. Calculate Initial Population: To find the population in 17901790, we need to calculate PP for t=0t=0, since 17901790 is the starting point of our model.\newlineWe use the equation P=4000(1.031)tP=4000(1.031)^{t} and substitute tt with 00.\newlineP=4000(1.031)(0)P = 4000(1.031)^{(0)}\newlineSince any number raised to the power of 00 is 11, we have:\newlineP=4000×1P = 4000 \times 1\newlineP=4000P = 4000
  2. Calculate Population in 18601860: To find the population in 18601860, we need to calculate PP for t=70t=70, since 18601860 is 7070 years after 17901790.\newlineWe use the equation P=4000(1.031)tP=4000(1.031)^{t} and substitute tt with 7070.\newlineP=4000(1.031)70P = 4000(1.031)^{70}\newlineNow we calculate (1.031)70(1.031)^{70} using a calculator.\newline(1.031)706.144(1.031)^{70} \approx 6.144\newlineSo, P4000×6.144P \approx 4000 \times 6.144\newlineP24576P \approx 24576

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