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What is the general formula for exponential growth of a population?

Answer 1

Isabella Ackerman

Population #[P]= Ce^[kt#

If the rate of growth #P# is proportional to itself, then with respect to time #t#,

#[dP]/dt=kP#, ....inverting both sides, .....#dt/[dP]=[1]/[kP# and so integrating both sides

#intdt=int[dP]/[kP#, thus,..... #t=1/klnP +# a constant............#[1]#

Suppose #P# is some value # C# when# t=0#, substituting

#0=1/klnC+# constant, therefore the constant #= -1/klnC# and so substituting this value for the constant in ...#[1]# we have ,

#t= 1/k[ln P-lnC]# = #1/k ln[P/C]#, therefore , #kt=ln[p/C]#[ theory of logs] and so

#e^[kt]=P/C#......giving # P=Ce^[kt#. The constant #k# will represent the excess of births over deaths or vice versa for a decreasing rate.

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Answer 2

Isaiah Allen

The general formula for exponential growth of a population is given by:

[ P(t) = P_0 \times e^{rt} ]

Where:

( P(t) ) is the population size at time ( t ),
( P_0 ) is the initial population size (at ( t = 0 )),
( r ) is the exponential growth rate, and
( e ) is the base of the natural logarithm (approximately equal to 2.71828).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.