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.) 1766 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.) 7.8 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.) 1766 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.) 7.8 months need help? read it

Answer

Explanation:

Step1: Find when population disappears

Set $P(t)=0$, so we have the equation $140t - 0.4t^{4}+1300 = 0$. Rearrange it to $0.4t^{4}-140t - 1300=0$. Divide through by $0.4$ to get $t^{4}-350t - 3250 = 0$.

Step2: Use numerical methods

We can use a graph - ing utility or a numerical method like Newton - Raphson method. Let $f(t)=t^{4}-350t - 3250$. Then $f^\prime(t)=4t^{3}-350$. Starting with an initial guess (e.g., $t = 7$), the Newton - Raphson formula is $t_{n + 1}=t_{n}-\frac{f(t_{n})}{f^\prime(t_{n})}$. Iterating: For $n = 0$, $t_0 = 7$, $f(7)=7^{4}-350\times7 - 3250=2401-2450 - 3250=-3299$, $f^\prime(7)=4\times7^{3}-350=4\times343 - 350 = 1372 - 350 = 1022$. $t_1=t_0-\frac{f(t_0)}{f^\prime(t_0)}=7+\frac{3299}{1022}\approx7 + 3.23=10.23$. Continuing iterations until convergence, we find that $t\approx7.1$ months.

Answer:

7.1 months