Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

A bird searches bushes in a field for insects. The total weight of insects found after \(t\) minutes of searching a single bush is given by \(w(t)=\frac{2 t}{t+4}\) grams. Draw a graph of \(w .\) From your graph, does it appear that a bird should search a single bush for more than 10 minutes? It takes the bird one minute to move from one bush to another. How long should the bird search each bush in order to harvest the most insects in an hour of feeding?

Step-by-Step Solution

Answer

The bird should search each bush for 9 minutes to maximize insect harvest in an hour.

Step 1: Understanding the Function

The function given is \( w(t) = \frac{2t}{t+4} \). It describes the weight of insects found by a bird after searching for \( t \) minutes in a single bush.

Step 2: Analyzing the Behavior of the Function

To understand how the function behaves, we should consider what happens as time \( t \) increases. For very large \( t \), the function approaches \( w(t) = 2 \), suggesting a maximum possible weight of insects found.

Step 3: Graphing the Function

Graph \( w(t) = \frac{2t}{t+4} \) for \( t \) ranging from 0 to 10. Notice the function is increasing rapidly at first and then starts to converge towards 2 as \( t \) increases beyond 10.

Step 4: Determining Optimal Search Time for a Single Bush

Consider the weight of insects found in more than 10 minutes. Since \( w(t) \) increases slowly after \( t = 10 \), searching beyond 10 minutes in a single bush provides diminishing returns.

Step 5: Calculating Total Insects Harvested in an Hour

The bird takes 1 minute to move to a new bush, meaning it spends a total of \( t + 1 \) minutes per bush. To maximize the total insects found in 60 minutes, we set up the inequality: \( \frac{60}{t+1} \cdot \frac{2t}{t+4} \). Use different integer values for \( t \) to approximate the highest quantity of insects.

Step 6: Optimizing Using Trial for Values of t

Test different values like \( t = 5 \), \( t = 6 \), ..., up to \( t = 10 \). Calculate the total harvest for each and find that \( t = 9 \) gives the highest result without exceding 60 minutes, as it optimally balances search and travel time.

Key Concepts

Function Behavior Analysis

To delve into function behavior analysis, start by understanding the function given in the exercise, which is \( w(t) = \frac{2t}{t+4} \). This function models the weight of insects found by a bird while searching a bush for \( t \) minutes. Recognizing the behavior of the function over different time intervals gives us insights into its nature.
As \( t \) increases, particularly when times become very large or small, the function tends to reveal significant aspects of its behavior. For instance, when \( t \) is very large, the function \( w(t) \) approximates to 2, indicating that this is the upper bound of the weight of insects the bird might find. This horizontal asymptote at \( w = 2 \) suggests that the bird gains less additional weight as time continues to increase.

Understanding such behavior helps guide decisions about optimizing the bird's search time for maximum efficiency. By focusing on the rapid increase in weight and identifying where gains slow significantly, we can better plan the bird's search strategy.

Graphing Functions

Graphing the function \( w(t) = \frac{2t}{t+4} \) reveals a lot about its dynamics. Drawing a graph as suggested for \( t \) ranging from 0 to 10 helps visually assess how the bird’s efforts translate into returns. Initially, the graph shows a steep incline, meaning that the first few minutes of searching are highly productive.

• **Initial Increase**: The graph rises sharply, indicating that each minute yields a significant amount of insects.
• **Plateau Phase**: After a point, particularly noticeable after \( t = 10 \), the graph begins to level off. This plateau suggests diminishing returns, as additional time contributes only marginally to the insect weight.

The value \( w(t) \) closing towards 2 forms a visual guidepost; the slow rise indicates less value in extended searches. Using such insights, a bird searching beyond 10 minutes gains little extra benefit, which is crucial for optimizing its efficiency over longer periods.

Optimization Strategy

When devising an optimization strategy, consider both collecting insects and moving between bushes. With the function \( w(t) = \frac{2t}{t+4} \), the bird wants to find the optimal balance between searching time in one bush and the transition to another. This process aligns with maximizing the total weight of insects within a set period, such as an hour.

Calculate using the expression \( \frac{60}{t+1}\cdot\frac{2t}{t+4} \), which incorporates both search and transfer time. Try different values of \( t \) to approximate the optimal outcome. For instance:

• Shorter times (e.g., \( t = 5 \)) reflect frequent moves, which might decrease overall efficiency because higher weight per bush isn’t leveraged fully.
• Extending to \( t = 9 \) results in a better harvest by optimizing both capture and transition processes over the available time.

Ultimately, the best strategy identified is when \( t = 9 \), providing the best compromise of minimized return trips and maximized collection. This approach highlights the importance of strategic planning in problem-solving scenarios involving both time and resource constraints.