Question

 species of animal is introduced to a small island. Suppose the popula- tion of the species is represented by P(t) = –4 + 9t2 + 400, where t is the time in years. Determine when the population becomes zero. Explain each step.​

Answer 1

CORRECT QUESTION

A species of animal is introduced to a small island. Suppose the population of the species is represented by

$\sf{P(t) = - {t}^{4} + 9 {t}^{2} + 400}$

where t is the time in years. Determine when the population becomes zero. Explain each step

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

Here the population of the species is represented by

$\sf{P(t) = - {t}^{4} + 9 {t}^{2} + 400}$

where t is the time in years

where t is the time in yearsNow in order to determine time when the population becomes zero we put

$\sf{P(t) = 0}$

$\implies \: \sf{ - {t}^{4} + 9 {t}^{2} + 400 = 0}$

$\implies \: \sf{ {t}^{4} - 9 {t}^{2} - 400 = 0}$

$\implies \: \sf{ {t}^{4} - 25 {t}^{2} +16 {t}^{2} - 400 = 0}$

$\implies \: \sf{ {t}^{2} ( {t}^{2} - 25) +16( {t}^{2} - 25) = 0}$

$\implies \: \sf{ ( {t}^{2} - 25) ( {t}^{2} + 16) = 0}$

$\sf{ So \: either} \: \: \sf{ ( {t}^{2} - 25) = 0 \: \: \: \: or \: \: \: ( {t}^{2} + 16) = 0}$

$\sf{ So \: either} \: \: \sf{ {t}^{2} = 25\: \: \: \: or \: \: \: {t}^{2} = - 16}$

Since time ( t ) is a real number

$\sf{ So \: \: \: \: {t}^{2} \ne - 16}$

Hence

$\sf{ So \:} \: \: \sf{ {t}^{2} = 25\: \: \: \:}$

$\implies \: \sf{ {t}^{} = \pm \: 5\: \: \: \:}$

Since time ( t ) can't be negative

Hence

$\sf\sf{ {t} = 5\: \: \: \:}$

RESULT

After 5 years the population becomes zero

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