The rabbit population on a small island is observed to be...

The rabbit population on a small island is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where $t$ is the time (in months) since observations of the island began. a. When is the maximum population attained, and what is that maximum population? b. When does the rabbit population disappear from the island?

The rabbit population on a small island is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where $t$ is the time (in months) since observations of the island began.
a. When is the maximum population attained, and what is that maximum population?
b. When does the rabbit population disappear from the island?

Key Concepts

Derivative and Critical Points

In calculus, derivatives are used to find the critical points of a function by setting the derivative equal to zero. These points may indicate where a function attains its maximum or minimum value. Understanding derivative tests is essential in optimizing functions, especially when determining optimal or extreme values in modeling scenarios.

Optimization

Optimization is the process of finding the best value, such as the maximum or minimum of a function, within a given context. In many real-world applications, such as determining the maximum population in a model, identifying and analyzing these extreme points is crucial for making informed predictions and decisions.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole number exponents with constant coefficients. Their well-understood behavior makes them suitable for modeling a wide range of natural phenomena, including population dynamics. Analyzing their properties, such as turning points and end behavior, provides insights into long-term trends and behaviors.

Solving Equations to Find Zeros

Finding the zeros of a function involves setting the function equal to zero and solving for the variable. In practical contexts, such as population models, zeros can represent the points in time when a population reaches a critical threshold, such as extinction or disappearance. This concept is fundamental in various applications that require determining when a quantity falls to a particular value.

Transcript

00:03 So in this question it is given to me, et is equal to 120t minus 0 .4 t.

00:12 Plus 1000, where the time t is in months and the et denotes the rabbit population.

00:26 Alright.

00:28 So in some island or something it denotes the rabbit population.

00:34 Now to find out the maximum population attained, we have to understand the graph and from the graph we can comment everything.

00:43 So let us see the graph of this given function quickly.

00:48 So this is how i have the graph.

00:51 This is the x -axis and this corresponds to the y -axis which denotes p t that is a population and x -axis denotes the time t that is in months.

01:05 So if i have to understand that the maximum population is attained at the point where it is attaining its maxima.

01:14 So this is the point where it is attaining its maxima.

01:17 On careful inspection, the coordinates will be obtained as 4 .23 .13 .139.

01:27 So this is the point of maximum of the graph.

01:30 What does it imply? this implies that the maximum population is the y -body net that is 1 ,379 .5 mathematically, but in reality it can be considered as 1379 can be considered as the maximum population.

01:59 So the maximum population is 1379 rabbits.

02:06 Now i have to tell which occurs in time, this represents the time...

Please give Ace some feedback