Problem 39 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

It is estimated that the weekly output at a certain plant is \(Q(x)=50 x^{2}+9,000 x\) units, where \(x\) is the number of workers employed at the plant. Currently there are 30 workers employed at the plant. a. Use calculus to estimate the change in the weekly output that will result from the addition of 1 worker to the force. b. Compute the actual change in output that will result from the addition of 1 worker.

Step-by-Step Solution

Answer

Estimated change: 12,000 units, Actual change: 12,050 units.

Step 1: Determine the function for weekly output

The weekly output function is given as \( Q(x) = 50x^2 + 9,000x \).

Step 2: Find the derivative of the function

To estimate the change in weekly output, determine the derivative of \( Q(x) \). The derivative represents the rate of change of the output with respect to the number of workers: \[ Q'(x) = \frac{d}{dx} (50x^2 + 9,000x) \].

Step 3: Calculate the derivative

Apply the power rule for differentiation: \( Q'(x) = 100x + 9,000 \).

Step 4: Evaluate the derivative at the current number of workers

Substitute \( x = 30 \) into \( Q'(x) \): \[ Q'(30) = 100(30) + 9,000 = 12,000 \].

Step 5: Estimate the change in output

Using the derivative, the estimated change in output when adding one worker (from 30 to 31 workers) is: \( Q'(30) \times \Delta x = 12,000 \times 1 = 12,000 \) units.

Step 6: Calculate the actual change in output

Compute the actual weekly output for 31 workers and 30 workers using \( Q(x) \): \[ Q(31) = 50(31)^2 + 9,000(31) \] and \[ Q(30) = 50(30)^2 + 9,000(30) \].

Step 7: Evaluate the outputs at 31 and 30 workers

\[ Q(31) = 50(31)^2 + 9,000(31) = 50(961) + 279,000 = 48,050 + 279,000 = 327,050 \] units, \[ Q(30) = 50(30)^2 + 9,000(30) = 50(900) + 270,000 = 45,000 + 270,000 = 315,000 \] units.

Step 8: Find the actual change in weekly output

Calculate the difference between the outputs at 31 and 30 workers: \[ Q(31) - Q(30) = 327,050 - 315,000 = 12,050 \] units.

Key Concepts

derivatives

The derivative in calculus is a powerful tool to understand how a function changes as its input changes. In our case, we use the derivative of the weekly output function to estimate how the output will change if we alter the number of workers. The weekly output function is given by \(Q(x) = 50x^2 + 9,000x\). To find the derivative, we apply the differentiation rules: \[ Q'(x) = 100x + 9,000 \]. This formula captures the 'instantaneous rate of change' of the output at any given number of workers. By evaluating this at specific values, we can predict output changes without recalculating complex expressions. For instance, when \(x=30\), \( Q'(30) = 12,000 \), indicating a steep increase.

rate of change

The rate of change tells us how one quantity changes in relation to another. In our problem, the rate of change is the derivative \(Q'(x)\) of the output function. It tells us how the output changes as more workers are employed. This rate helps businesses make informed decisions. For example, we found that when \(x=30\) workers, the rate of change is 12,000 units per worker. This numerical value helps predict the increase in output for each additional worker, providing insights for workforce adjustments.

output function

The output function, \(Q(x) = 50x^2 + 9,000x\), models the total production based on the number of workers. It's essential for understanding how employment size impacts productivity. This specific quadratic function shows an increasing trend; both terms \(50x^2\) and \(9,000x\) contribute to the total output. Companies use such functions to forecast production levels. Evaluating it directly, we see that 30 workers produce 315,000 units. Increasing to 31 workers boosts output to 327,050 units. Thus, small changes in \(x\) significantly affect the overall output.

marginal analysis

Marginal analysis focuses on the impact of adding or reducing one unit of input. It's a crucial concept in economics and business. In our scenario, it involves analyzing the output change due to one more worker. By using the derivative \(Q'(x)\), we estimated the marginal output at \(Q'(30) = 12,000\) units. Performing actual calculations, we found the change to be 12,050 units. Marginal analysis helps determine the optimal input levels for maximizing profit and efficiency. Here, it highlights the benefits of adding each extra worker and guides decisions for staffing.