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Numerade Educator

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Problem 10 Medium Difficulty

(a) Assume that the carrying capacity for the US population is 800 million. Use it and the fact that the population was 282 million in 2000 to formulate a logistic model for the US population.
(b) Determine the value of $ k $ in your model by using the fact that the population in 2010 was 309 million.
(c) Use your model to predict the US population in the years 2100 and 2200.
(d) Use your model to predict the year in which the US population will exceed 500 million.

Answer

a) $P(t)=\frac{800}{1+1.836879 e^{-k t}}$
b) $k \approx 0.0145$
c) Population in the year 2100 will be 559 million
Population in the year 2200 will be 727.57 million
d) Population in the year 2077 will be 500 million

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Video Transcript

his clear so new Marine here. So for part A, we have, um, is equal to 800 on. We have p of zero vehicle 2 to 82. So hee value is equal to 800 minus 2 82 over to 82 which is about 1.83 six eight 79 So we got pft this equal to 800 over one plus 1.83 6879 e to the negative, Katie, for a part, viewing our equation. So we're given p of 10 as equal to 309. So it's 309 is equal to 800 all over one plus 1.836 879 e to the negative 10 k and then we end up getting e. The 10 k is equal to 1.156 We take the natural log when we get Katie to be about 0.145 for part C, we have the equation. P F T is equal to 800 over one plus 800 over one plus 1.836 879 e. It's the negative 0.145 t well, for 2100 Will BP of 100 which is about 559 million. Her p of 200. We'll be about 729. 27 wait 57 million for part B. We have our equation that we used right here for part C when we are finding the ear when the population is gonna be 500 million. So we just made that equation be equal to 500 all over one plus 1.83 6879 no eat the native 0.145 t We'll continue over here and then we get eat the zero point there are 145 T is equal to five times 1.836 879 over three when we get a T value of about 77