a graphing device is recommended. the rabbit population on a small island is observed to be given by the…

a graphing device is recommended. the rabbit population on a small island is observed to be given by the function p(t)=140t - 0.4t^4+1,300 where t is the time (in months) since observations of the island began. (a) what is the maximum population attained? (round your answer to the nearest whole number.) 1765 × rabbits when (in months) is the maximum population? (round your answer to one decimal place.) 4.4 √ months (b) when (in months) does the rabbit population disappear from the island? (round your answer to one decimal place.) 23.2 × months need help? read it

a graphing device is recommended. the rabbit population on a small island is observed to be given by the function p(t)=140t - 0.4t^4+1,300 where t is the time (in months) since observations of the island began. (a) what is the maximum population attained? (round your answer to the nearest whole number.) 1765 × rabbits when (in months) is the maximum population? (round your answer to one decimal place.) 4.4 √ months (b) when (in months) does the rabbit population disappear from the island? (round your answer to one decimal place.) 23.2 × months need help? read it

Answer

Explanation:

Step1: Find the derivative of $P(t)$

Differentiate $P(t)=140t - 0.4t^{4}+1300$ using power - rule. $P^\prime(t)=140 - 1.6t^{3}$.

Step2: Set the derivative equal to zero

To find critical points, set $P^\prime(t) = 0$. So, $140 - 1.6t^{3}=0$. Then $1.6t^{3}=140$, and $t^{3}=\frac{140}{1.6}=87.5$. Thus, $t=\sqrt[3]{87.5}\approx4.4$ months.

Step3: Find the maximum population

Substitute $t = 4.4$ into $P(t)$. $P(4.4)=140\times4.4-0.4\times(4.4)^{4}+1300$. $P(4.4)=616-0.4\times374.8096 + 1300$. $P(4.4)=616 - 149.92384+1300$. $P(4.4)\approx1766$ rabbits.

Step4: Find when the population disappears

Set $P(t)=0$, so $140t - 0.4t^{4}+1300 = 0$. Rearrange to $0.4t^{4}-140t - 1300 = 0$. Divide by $0.4$ to get $t^{4}-350t - 3250=0$. Using a graphing device or numerical methods (such as Newton - Raphson method), we find the positive root of the equation. The root is approximately $t\approx7.9$ months.

Answer:

(a) 1766 rabbits (b) 7.9 months