Problem 26 - Applied Calculus

Question

In a small printing business, \(P=2 N^{0.6} V^{0.4},\) where \(N\) is the number of workers. \(V\) is the value of the equipment, and \(P\) is production, in thousands of pages per day. (a) If this company has a labor force of 300 workers and 200 units worth of equipment, what is production? (b) If the labor force is doubled (to 600 workers), how does production change? (c) If the company purchases enough equipment to double the value of its equipment (to 400 units), how does production change? (d) If both \(N\) and \(V\) are doubled from the values given in part (a), how does production change?

Step-by-Step Solution

Answer

(a) 436.27; (b) 740.08; (c) 695.86; (d) 1179.46 thousand pages per day.

Step 1: Understand the formula for production

We are given the formula for production, which is \(P = 2 N^{0.6} V^{0.4}\). The variables are: \(N\), the number of workers, and \(V\), the value of the equipment. \(P\) represents production in thousands of pages per day.

Step 2: Calculate production with current values (part a)

Substitute \(N = 300\) and \(V = 200\) into the production formula:\[ P = 2 \times 300^{0.6} \times 200^{0.4} \]Calculate each power:\[ 300^{0.6} \approx 34.5550 \]\[ 200^{0.4} \approx 6.3096 \]Then, calculate \(P\):\[ P = 2 \times 34.5550 \times 6.3096 \approx 436.27 \]Thus, production is approximately 436.27 thousand pages per day.

Step 3: Doubling the workforce (part b)

Now, double the workforce to \(N = 600\) while keeping \(V = 200\) constant.Substitute into the formula:\[ P = 2 \times 600^{0.6} \times 200^{0.4} \]Calculate:\[ 600^{0.6} \approx 58.6344 \]\[ 200^{0.4} \approx 6.3096 \]Thus, \(P\) is:\[ P = 2 \times 58.6344 \times 6.3096 \approx 740.08 \]Production increases to approximately 740.08 thousand pages per day.

Step 4: Doubling the equipment value (part c)

Keep the workforce at \(N = 300\) and double the equipment value to \(V = 400\).Substitute into the formula:\[ P = 2 \times 300^{0.6} \times 400^{0.4} \]Calculate:\[ 300^{0.6} \approx 34.5550 \]\[ 400^{0.4} \approx 10.0794 \]Thus, \(P\) is:\[ P = 2 \times 34.5550 \times 10.0794 \approx 695.86 \]Production increases to approximately 695.86 thousand pages per day.

Step 5: Doubling both workforce and equipment value (part d)

Now, double both the workforce and the equipment value: \(N = 600\) and \(V = 400\).Substitute these values into the formula:\[ P = 2 \times 600^{0.6} \times 400^{0.4} \]Calculate:\[ 600^{0.6} \approx 58.6344 \]\[ 400^{0.4} \approx 10.0794 \]Thus, \(P\) is:\[ P = 2 \times 58.6344 \times 10.0794 \approx 1179.46 \]When both the workforce and equipment value are doubled, production increases to approximately 1179.46 thousand pages per day.

Key Concepts

Production Function

In applied calculus and economics, a production function is a mathematical model representing the relationship between inputs and outputs. It demonstrates how varying quantities of inputs result in different levels of production. In our scenario, the production function is described by the formula: \[ P = 2 N^{0.6} V^{0.4} \]Here, \(P\) is the output, specifically the number of thousands of pages printed per day. \(N\) is the number of workers, and \(V\) signifies the value of the equipment. Each input is raised to a power, reflecting the elasticity or responsiveness of production to changes in each input.

• The coefficient "2" represents a base level of productivity constant for this setup.
• Exponents 0.6 and 0.4 indicate the contribution or weight of each input toward the production.

This function helps businesses understand how changes in resources like labor and equipment affect output levels.

Exponents and Powers

Exponents and powers are used to express how many times a number, called the base, is multiplied by itself. In the given production function, the expressions \(N^{0.6}\) and \(V^{0.4}\) signify the use of exponents to model non-linear growth.

• \(N^{0.6}\) means the number of workers' contribution to production isn't directly proportional but rather increases at a decreasing rate.
• \(V^{0.4}\) shows how the equipment value helps increase production, but like labor, it is subject to diminishing returns.

Understanding these powers is crucial for predicting how changes in inputs impact production. The exponents reflect real-world cases where doubling an input doesn't necessarily double the output due to factors like worker motivation and equipment utilization efficiency.

Business Applications

The production function has numerous applications in business, providing insights into efficient resource allocation and planning growth strategies. By analyzing how inputs affect outputs, businesses can make data-driven decisions to optimize production. - **Resource Allocation**: Adjusting the number of workers or upgrading equipment strategically impacts production levels, allowing for cost-effective expansion. - **Budgeting**: Understanding the diminishing returns seen in exponents (e.g., 0.6, 0.4) assists in budgeting, as it lets managers know the point beyond which increasing inputs yields minimal output improvements. - **Scenario Analysis**: As illustrated, businesses can simulate various scenarios, such as doubling workforce or equipment, to forecast production changes and plan accordingly. These applications underscore how calculus concepts like production functions optimize business operations, ultimately leading towards more robust and insightful strategic planning.