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A given field mouse population satisfies the differential equation dp dt = 0.5p − 410 where p is the number of mice and t is the time in months. (a) Find the time at which the population becomes extinct if p(0) = 770. (Round your answer to two decimal places.) 25 Incorrect: Your answer is incorrect. month(s) (b) Find the time of extinction if p(0) = p0, where 0 < p0 < 820. Incorrect: Your answer is incorrect. month(s) (c) Find the initial population p0 if the population is to become extinct in 1 year. (Round your answer to the nearest integer.) p0 = mice Additional Materials
A given field mouse population satisfies the differential equation dp dt = 0.5p − 410 where p is the number of mice and t is the time in months. (a) Find the time at which the population becomes extinct if p(0) = 770. (Round your answer to two decimal places.) 25 Incorrect: Your answer is incorrect. month(s) (b) Find the time of extinction if p(0) = p0, where 0 < p0 < 820. Incorrect: Your answer is incorrect. month(s) (c) Find the initial population p0 if the population is to become extinct in 1 year. (Round your answer to the nearest integer.) p0 = mice Additional Materials
For the given case of increment ofmice'population, we get following figures:After 5.59monthsapprox, thepopulationofmicewillextinct.Theextinction time(in months) ofpopulationofmicewhen its given thatis given by:Theinitial populationofmicefor given conditions would be approx 818What is differential equation?Anequationcontainingderivativesof avariablewithrespectto some other variable quantity is called differential equations. The derivatives might be of any order, some terms might containproductofderivativesand thevariableitself, or withderivativesthemselves. They can also be formultiple variables.For the considered case, thepopulationofmicewith respect totimepassed inmonthsis given by thedifferential equation:Taking samevariableterms on same side, and then integrating, we get:where C₁ isintegration constant.Since it is specified that attimet = 0, thepopulationp = 770, therefore,puttingthesevaluesin theequationobtained above, we get:Therefore, we get the relation between p and t as:Calculating the needed figures for each sub-parts of the problem:a): Thetimeat which thepopulationbecomesextinct.Let it be t at which p becomes 0, then, from theequationobtained, we get:Thus, after 5.59monthsapprox, thepopulationofmicewillextinct.b) Find thetimeofextinctionif p(0) = p0, where 0 < p0 < 820From the equationputtingwhen t = 0, we get the value of C as:Thus, theequationbecomesAttimeofextensiont months, p becomes 0, thus,Thus, theextinction time(in months) ofpopulationofmicewhen its given thatis given by:c) Find theinitial populationif thepopulationis to becomeextinctin 1 year.Putting t = 12 (since t is measured inmonths, and that 1 year = 12months) in theequationobtained in the second part, we get the value of initialpopulationas:Thus, theinitial populationofmicefor given conditions would be approx 818Therefore, for the given case of increment ofmice'population, we get following figures:After 5.59monthsapprox, thepopulationofmicewillextinct.Theextinction time(in months) ofpopulationofmicewhen its given thatis given by:Theinitial populationofmicefor given conditions would be approx 818Learn more aboutdifferential equationshere:brainly.com/question/14744969...