Problem 34 - Applied Calculus

Question

The energy expended by a bird per day, \(E,\) depends on the time spent foraging for food per day, \(F\) hours. Foraging for a shorter time requires better territory, which then requires more energy for its defense. \(^{3}\) Find the foraging time that minimizes energy expenditure if $$E=0.25 F+\frac{1.7}{F^{2}}$$

Step-by-Step Solution

Answer

The minimum energy expenditure occurs at approximately 2.38 hours of foraging.

Step 1: Define the Objective Function

The objective function given is the energy expended per day, \(E\), which is dependent on the foraging time, \(F\). The function is: \(E = 0.25F + \frac{1.7}{F^2}\). Our goal is to find the value of \(F\) that minimizes \(E\).

Step 2: Differentiate the Objective Function

To find the minimum energy expenditure, differentiate the function \(E = 0.25F + \frac{1.7}{F^2}\) with respect to \(F\). \[\frac{dE}{dF} = 0.25 - \frac{3.4}{F^3}\].

Step 3: Set Derivative to Zero

To find critical points, set the derivative equal to zero: \[0.25 - \frac{3.4}{F^3} = 0\]. Solve for \(F\).

Step 4: Solve the Equation

The equation \(0.25 = \frac{3.4}{F^3}\) can be rearranged to \(F^3 = \frac{3.4}{0.25}\). Solution of the equation leads to: \[F = \sqrt[3]{13.6} \approx 2.38\].

Step 5: Confirm the Minimum with Second Derivative Test

Calculate the second derivative \(\frac{d^2E}{dF^2} = \frac{10.2}{F^4}\). Since \(\frac{10.2}{F^4}\) is positive for \(F > 0\), \(E\) is concave up, confirming \(F = 2.38\) is a minimum.

Key Concepts

Objective Function

In optimization problems, the objective function is a vital element that determines what is being optimized. Here, the objective function is the energy expended by the bird per day, denoted as \( E \). It depends on the time spent foraging for food, expressed as \( F \) hours.
The function is given by \( E = 0.25F + \frac{1.7}{F^2} \). This equation suggests that energy expenditure increases linearly with foraging time, \( F \), and also includes an inverse cubic term \( \frac{1.7}{F^2} \) attributed to territorial defense. Our task in this scenario is to find the optimal \( F \) that minimizes the energy \( E \).
By analyzing and altering the objective function, we can better understand and refine the conditions under which the bird expends the least energy.

Derivative Calculation

Calculating the derivative of the objective function is a crucial step in optimization. It helps to determine the rate of change of energy expenditure with changes in foraging time. To optimize \( E \), we need to find its derivative with respect to \( F \).
The original function \( E = 0.25F + \frac{1.7}{F^2} \) is differentiated to produce:
\[\frac{dE}{dF} = 0.25 - \frac{3.4}{F^3}\]
This derivative indicates the slope of the energy function at any given \( F \). A slope of zero at some point \( F \) suggests a potential minimum or maximum, leading us to the next step in our analysis.

Critical Points

Critical points in calculus are values of \( F \) where the derivative \( \frac{dE}{dF} = 0 \). To find these points, we set the derivative computed earlier to zero:
\[0.25 - \frac{3.4}{F^3} = 0\]
Solving gives us \( F^3 = \frac{3.4}{0.25} \), providing the critical point \( F = \sqrt[3]{13.6} \approx 2.38 \).
This critical point is the candidate for where the minimum energy expenditure occurs. It is essential to confirm that this critical point indeed represents a minimum by further analysis.

Second Derivative Test

To confirm whether the critical point found provides a minimum, we use the second derivative test. The second derivative \( \frac{d^2E}{dF^2} \) assesses the concavity of the function:
\[\frac{d^2E}{dF^2} = \frac{10.2}{F^4}\]
Since \( \frac{10.2}{F^4} \) is positive for all \( F > 0 \), the energy function is concave up at \( F = 2.38 \), indicating a local minimum. Thus, this test confirms that a foraging time of approximately 2.38 hours minimizes the bird's daily energy expenditure.
This concludes the optimization procedure, providing an analytical approach to finding optimal conditions efficiently.