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A population grows according to the given logistic equation, where $ t $ is measured in weeks.(a) What is the carrying capacity? What is the value of $ k? $(b) Write the solution of the equation.(c) What is the population after 10 weeks?$ \frac {dP}{dt} = 0.02P - 0.0004P^2, P(0) = 40 $

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 4

Models for Population Growth

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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A population grows accordi…

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$1-2$ A population grows a…

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The logistic equation mode…

02:52

Logistic growth: $P(t)=\fr…

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The logistic differential …

hey is clear. So when you right here. So we're given the DP over. DT was equal to zero point zero to pee minus zero point 00 Thorough for peace where we got factor out A 0.2 p, you know, one minus 0.2 and this is equal to 0.2 p times one minus p over 50 for part B. We know that the carrying capacity is gonna be part is gonna be 50 part, eh? And we know K is equal to point 02 So we know that M is equal to 50 and p of zero is equal to 40. So for our A value, which is equal to 50 minus 40 over 40 which is equal to point 25 So our solution is P of tea is equal to 50 over one plus 0.25 eats the negative 0.2 tea for part C. We have the equation from part B. So we're looking at the population 10 weeks after, so we just plug in p of tea and this is around 41

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