Problem 40 - Calculus with Concepts in Calculus

Question

A farmer wishes to employ tomato pickers to harvest 62,500 tomatoes. Each picker can harvest 625 tomatoes per hour and is paid \(\$ 6\) per hour. In addition, the farmer must pay a supervisor \(\$ 10\) per hour and pay the union \(\$ 10\) for each picker employed. a. How many pickers should the farmer employ to minimize the cost of harvesting the tomatoes? b. What is the minimum cost to the farmer?

Step-by-Step Solution

Answer

Employ 1 picker for a minimum cost of $2600.

Step 1: Determine Total Hours Needed

Calculate the total number of hours required to harvest the 62,500 tomatoes. Each picker can harvest 625 tomatoes per hour. So, divide the total number of tomatoes by the number of tomatoes each picker can harvest per hour.divide \(62,500\) (total tomatoes) by \(625\) (tomatoes per picker per hour) to find the total hours needed:\[\text{Total Hours Needed} = \frac{62,500}{625} = 100\text{ hours}\]

Step 2: Express Total Cost as Function of Pickers

The cost involves payments to both the pickers and the supervisor. If \(x\) is the number of pickers, then the cost per hour for all pickers is \(6x\) dollars, the supervisor cost per hour is \(10\) dollars, and the union fee per picker is \(10x\) dollars.The total hourly cost function is:\[\text{Total Cost per hour} = 6x + 10 + 10x = 16x + 10\]

Step 3: Determine Total Cost for Harvesting Period

Calculate the total cost to harvest all tomatoes by multiplying the cost per hour by the total hours needed (from Step 1).With \(100\) hours of work needed:\[\text{Total Cost} = (16x + 10) \times 100 = 1600x + 1000\]

Step 4: Minimize the Total Cost Function

Identify the number of pickers, \(x\), that minimizes the total cost function. Since it's a linear function with respect to \(x\), the smaller the value of \(x\), the lower the cost. Therefore, you should employ a minimum number of pickers that ensures all tomatoes are harvested within the 100-hour timeframe.

Step 5: Find Minimum Number of Pickers

Each picker can work up to 100 hours, so the minimum number of pickers is determined by ensuring all 100 hours are covered. Employing 1 picker would require \(100\) hours, which is the minimum, making it feasible to use as few as 1 picker.

Step 6: Calculate Minimum Cost

Substitute the minimum number of pickers (1 picker) into the total cost function.\[\text{Minimum Total Cost} = 1600 \times 1 + 1000 = 2600\]

Key Concepts

Linear Cost Functions

Understanding linear cost functions is essential in optimizing costs, especially in scenarios like agriculture where resources must be allocated efficiently. A linear cost function is a mathematical expression that describes how total cost changes based on some variable amount, often the quantity of resources used. In this exercise, the farmer’s cost for hiring pickers can be expressed as a linear function based on the number of pickers employed.The total cost per hour for the farmer includes payments to both the pickers and a supervisor. By defining the number of pickers as \( x \), we express the pickers' cost as \( 6x \) (since each picker earns \\(6 per hour), the supervisor's cost as a flat \\)10 per hour, and a union fee of \( 10x \) dollars. Thus, the hourly cost function can be simplified to:

• \( 16x + 10 \)

This linear expression allows us to see how the cost changes directly with the number of pickers. Linear cost functions like this make it easy to calculate total costs when fluctuating the number of resources, like workers, to find cost-effective solutions.

Resource Allocation

Resource allocation is about determining the best way to distribute resources to achieve a specific goal in the most efficient manner. In the context of the farmer's problem, resource allocation involves deciding how many pickers to hire to minimize the overall cost of harvesting tomatoes. The main challenge here is balancing the cost of employing the pickers with other fixed costs, such as the supervisor's wages and the union fees. The farmer needs to allocate the right number of hours each picker will work to optimize costs while ensuring the harvesting job is completed. In order to efficiently allocate resources:

• Assess the total hours needed (100 hours in this case) based on output per picker.
• Determine how many pickers are required to cover those hours.

By hiring the smallest number of pickers that meets the labor requirement, the farmer optimally allocates resources and minimizes expenses, which in this case is achieved by hiring only one picker for the full duration of 100 hours.

Problem Solving in Calculus

Problem solving using calculus involves techniques for optimizing solutions by finding maximum or minimum values for certain functions. Although the example provided doesn’t directly use calculus to find a derivative, it involves understanding function behavior to minimize costs.To analyze the total cost function derived earlier, we consider it a linear function of \( x \). However, in more complex scenarios, calculus would be applied as follows:

• Identify the function to be optimized, in this case, the cost function \( 1600x + 1000 \).
• Use calculus techniques to find points where the derivative is zero (these points often represent local maxima or minima).
• Evaluate these points to determine the absolute minimum or maximum points, which gives the optimal solution.

For linear functions like the one in this problem, straightforward reasoning lets us easily deduce that minimizing \( x \) at the smallest feasible integer (while meeting all conditions) provides the solution without needing calculus. This highlights how calculus aids in problem-solving by offering tools for analyzing more complex functions beyond simple linear equations.