Problem 82 - Algebra and Trigonometry

Question

Population Change 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? (GRAPH NOT COPY)

Step-by-Step Solution

Answer

The maximum population is attained at \( t = 75^{1/3} \) and the maximum population value is calculated at this \( t \). The rabbits disappear when \( P(t) = 0 \), which requires numerical methods to solve.

Step 1: Find the derivative

To determine when the maximum population is attained, we first find the derivative of the population function \( P(t) = 120t - 0.4t^4 + 1000 \). The derivative, \( P'(t) \), helps us find the critical points, which are candidates for maxima or minima. The derivative is \( P'(t) = 120 - 1.6t^3 \).

Step 2: Find critical points

Set the derivative equal to zero to find the critical points: \( 120 - 1.6t^3 = 0 \). Solving for \( t \), we get \( t^3 = 75 \), giving us \( t = 75^{1/3} \).

Step 3: Test for maxima

To confirm that this critical point is a maximum, use the second derivative test. Find \( P''(t) = -4.8t^2 \). Substitute \( t = 75^{1/3} \) into \( P''(t) \) to find \( P''(t) < 0 \), confirming a maximum at this point.

Step 4: Calculate maximum population

Substitute \( t = 75^{1/3} \) back into the original function \( P(t) = 120t - 0.4t^4 + 1000 \) to find the maximum population value. Calculate \( P(75^{1/3}) = 120(75^{1/3}) - 0.4(75) + 1000 \).

Step 5: Determine when population is zero

To find when the population disappears, solve the equation \( P(t) = 0 \): \( 120t - 0.4t^4 + 1000 = 0 \). This equation is a quartic and may require numerical methods or graphing to solve. Identify when \( P(t) = 0 \) using these methods.

Key Concepts

Critical Points

When dealing with functions and wanting to know where extremes occur, like maximums or minimums, we need to find critical points. These are points on a curve where the derivative equals zero or doesn't exist.
For population models, like the rabbit population on the island, knowing critical points helps identify when the population will reach its highest or lowest.

• Start by finding the derivative of the function.
• Set the derivative to zero to find potential critical points.
• Solve for the variable to identify them.

This helps in spotting where the highest or lowest numbers for populations can occur, an important aspect of understanding growth dynamics.

Derivative

A derivative in calculus represents the rate of change of a function concerning a variable. Think of it as a mathematical tool that tells us how something changes over time.
In our rabbit population problem, the function we have is dependent on time, so calculating the derivative helps us understand how quickly the population is changing at any given moment.

• For the function, \( P(t) = 120t - 0.4t^4 + 1000 \), we calculate the derivative:
• The derivative, \( P'(t) = 120 - 1.6t^3 \), shows changes in population growth.

By solving for when this derivative equals zero, we find points in time where the population stops increasing or decreasing, which are our critical points.

Second Derivative Test

The second derivative gives insight into the concavity of the function, or how the function is "bending" at its critical points. It determines whether these critical points are maxima, minima, or points of inflection.
Here's how it works:

• After finding critical points from the first derivative, calculate the second derivative.
• For our example, \( P''(t) = -4.8t^2 \).
• Plug in the critical points into the second derivative:

If \( P''(t) > 0 \), then the original function is concave up, indicating a local minimum. If \( P''(t) < 0 \), the function is concave down, indicating a local maximum. So, a negative result in our rabbit population problem confirms a maximum, telling us that the population reaches its peak.

Quartic Equation

A quartic equation is a polynomial of degree four. It can describe complex curves with multiple turning points, making them challenging to solve by simple algebra alone.
In many cases, using quartic equations requires multiple steps or even numerical methods for solutions. In our population modeling problem, we need this approach to find when the population will be zero. Here's an overview:

• Write the equation in standard form, like \( 120t - 0.4t^4 + 1000 = 0 \).
• This equation doesn't easily factor, so numerical solutions or graphing methods can help find roots.

Knowing how to approach quartic equations is important in applied settings, where they may describe growth, decay, or other natural phenomena.