Mathematics: Let P(t) be a population at time t. A simple population model supposes the rate of growth of the population is proportional to the population (a) Express the last sentence in the language of calculus. (b) Solve for P(t) in the equation you found in part (a) (c) Suppose the net birthrate of the population is .1, and the initial population is 100. (i) Find the population at time t 10.(ii) Find when the population is 1000.

Let P(t) be a population at time t. A simple population

Ответ:

KingOfLaBron23

29.01.2022

Mathematics

$(a) \dfrac{dP}{dt} =k P(t)\\(b)P(t)=Ce^{kt}$

(ii) $P(1000)\approx 26.88 \times 10^{44}\\$

Step-by-step explanation:

(a)The rate of growth of the population is proportional to the population, this is written as:

$\dfrac{dP}{dt} \propto P(t)\\$Introducing our proportionality constant, k\\ \dfrac{dP}{dt} =k P(t)$

(b)

$\dfrac{dP(t)}{P(t)} =k dt\\$Take the integral of both sides\\\int \dfrac{dP(t)}{P(t)} =\int k dt\\\ln P(t)=kt+C, $C a constant of integration\\Take the exponential of both sides\\e^{\ln P(t)}=e^{kt+C}\\P(t)=e^{kt}\cdot e^C $, (Since e^C$ is a constant, we then have:)\\P(t)=Ce^{kt}$

(c)

Suppose the net birthrate of the population is .1, and the initial population is 100.

k=0.1

P(0)=100

Substitution into P(t) gives:

$100=Ce^{kX0}$

C=100

Therefore:

$P(t)=100e^{0.1t}$

(i)When t=10

$P(10)=100e^{0.1 \times 10}\\=271.8\\\approx 272$

(ii)When t=1000

$P(1000)=100e^{0.1 \times 1000}\\=26.88 \times 10^{44}\\$

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Ответ:

kayli53531

12.05.2023

Mathematics

-0.91

Step-by-step explanation:

A linear relationship is said to be strong if the points on graph are close to a straight line ( correlation coefficient is nearest to 1) and weak if they widely scattered about the line ( correlation coefficient is nearest to -1) .

From the given graph it appeaser to be a strong negative correlation as the points are moving downwards diagonally close to a line which shows if one variable increases , the other decreases.

It means the correlation coefficient must be the nearest value to -1, i.e. -0.91.

Thus, the correlation coefficient most likely for the set of data shown =-0.91

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