Birth and death rates of animal populations typically arenot constant; instead, they vary periodically with the pas-sage of seasons. Find P(t) if the population P satisfiesthe differential equationdP= (k + b cos 27t)P,dtwhere t is in years and k and b are positive constants. Thusthe growth-rate function r(t) = k + b cos 2.rt varies peri-odically about its mean value k. Construct a graph thatcontrasts the growth of this population with one that hasthe same initial value Po but satisfies the natural growthequation P' = kP (same constant k). How would the twopopulations compare after the passage of many years?

Answered: Image /qna-images/answer/b5deb25e-068b-4206-9e90-9570c411c13c.jpg

Answered: Birth and death rates of animal…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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7. A commercial fishery is estimated to have carrying capacity 40 tons of salmon. Suppose the annual growth rate of total salmon population P (measured in tons) is governed by the logistic equation P' = P(1 – P/40) and initially there are 30 tons of fish. (i) What is the fish population after 1 year? (You may use e 2 2.72.) Suppose that the owner of the fishery decides to harvest h tons of salmon annually at a constant rate. Then (ii) ( What is the differential equation governing the fish population now? (iii) What is the maximun sustainable harvesting rate? (Justification is required.)
A 288 pound object is suspended from a spring. The spring stretches an extra 9 inches with the weight attached. The spring-and-mass system is then submerged in a viscous fluid that exerts 16 pounds of force when the mass has velocity 2 ft/sec. What is the differential equation for this spring-and-mass system? (Use u as the dependent variable.) This system is ? For what value of y would the system be critically damped? (if the given system is already critically damped, enter the given value for y.)
During the period from 1790 to 1930, a country's population P(t) (t in years) grew from 3.7 million to 118.6 million. Throughout this period, P(t) remained close to the solution of the initial value dP problem = 0.03145P-0.0001486P², P(0) = 3.7. dt (a) What 1930 population does this logistic equation predict? (b) What limiting population does it predict? (c) The country's population in 2000 was 350 million. Has this logistic equation continued since 1930 to accurately model the country's population?
Show that solving the logistic differential equation results in the logistic growth function
During the period from 1790 to 1920, a country's population P(t) (t in years) grew from 3.6 million to 101.6 million. Throughout this period, P(t) remained close to the solution of the initial value problem -0.0001494P² P(0) = 3.6. dP dt = 0.03136P-( (a) What 1920 population does this logistic equation predict? (b) What limiting population does it predict? (c) The country's population in 2000 was 259 million. Has this logistic equation continued since 1920 to accurately model the country's population? (a) The logistic equation predicts the population in 1920 to be million. (Do not round until the final answer. Then round to the nearest thousandth as needed.) K AI C H TU U →> C ¢ ➡ H i O + 2.4 e В → نه Y + - 8 .... : *** 5 Ø Co Less . N 0 .. H A E X U i 9 A C ש Σ VI 8 FR > in X
Suppose that P, is invested in a savings account in which interest is compounded continuously at 5.2% per year. That is, the balance P grows at the rate given by P = .052 P(t) dt a) Find the function P(t) that satisfies the equation. b) Suppose that $1200 is invested. What is the balance after 4 years? c) When will an investment of $1800 double itself?

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