Problem 115 - A Complete Resource Book in Mathematics for JEE Main

Question

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production \(P\) with respect to additional number of workers \(x\) is given by \(\frac{d P}{d x}=100-12 \sqrt{x}\). If the firm employs 25 more workers, then the new level of production of items is [2013] (A) 3000 (B) 3500 (C) 4500 (D) 2500

Step-by-Step Solution

Answer

The new production level is 3500 items, so the answer is (B) 3500.

Step 1: Understand the Problem

We have a production function where the rate of change, or derivative, of production \(P\) with respect to the number of additional workers \(x\) is \(\frac{dP}{dx} = 100 - 12\sqrt{x}\). We need to find the new production level after employing 25 more workers.

Step 2: Set Up the Integral

Determine the increase in production by integrating the rate of change \(\frac{dP}{dx} = 100 - 12\sqrt{x}\) from \(x = 0\) to \(x = 25\). The integral will be \(\int_{0}^{25} (100 - 12\sqrt{x}) \, dx\).

Step 3: Integrate the Function

Calculate the integral: \[ \int (100 - 12\sqrt{x}) \, dx = \int 100 \, dx - \int 12x^{1/2} \, dx. \]This gives: \[ 100x - 12 \cdot \frac{2}{3}x^{3/2} = 100x - 8x^{3/2}. \]We evaluate this from 0 to 25.

Step 4: Evaluate the Integral

Plug the limits of integration into the integrated function:\[ \left[100x - 8x^{3/2}\right]_{0}^{25} = \left(100(25) - 8(25)^{3/2}\right) - \left(100(0) - 8(0)^{3/2}\right). \]

Step 5: Simplify the Expression

Calculate the simplified expression:\[ 100(25) = 2500, \text{ and } 8(25)^{3/2} = 8 \times 125 = 1000. \]This results in:\[ 2500 - 1000 = 1500. \]

Step 6: Find the New Production Level

The initial production level is 2000 units. The increase in production is 1500 units. Therefore, the new production level is:\[ 2000 + 1500 = 3500. \]

Step 7: Choose the Correct Option

The answer choice corresponding to our calculated production level of 3500 is (B) 3500.

Key Concepts

Rate of Change

The rate of change is a fundamental concept in calculus that examines how a function's output changes as its input changes. In the context of this problem, we are looking at the change in production rate with respect to hiring additional workers. It is given by the derivative function \( \frac{dP}{dx} = 100 - 12\sqrt{x} \). This function tells us how production output is expected to change for each additional worker hired.

In practical terms, understanding the rate of change allows firms to predict how changes in the workforce can affect production. A positive rate indicates an increase in production as workers are added, while a negative rate may show decreasing returns. For the firm's decision-making, it's crucial to know the rate of change to efficiently allocate resources and maintain optimal production levels.

Production Function

The production function is a vital concept in economics and integral calculus that describes the relationship between input factors, like labor, and output, such as goods produced. In this problem, the production function isn't given explicitly, but its rate of change is. The function the students are dealing with specifies how production increases as more workers are employed.

By integrating the rate of change of the production function, we can determine the actual change in production level when the number of workers changes. This helps in calculating how employing additional workers impact production. The production function is a tool for businesses to understand efficiency, predict output levels, and make informed decisions about scaling operations.

Indefinite Integral

The indefinite integral, often known as the antiderivative, is the reverse process of differentiation. It helps us find the original function when we know its derivative. In this exercise, the indefinite integral was used to find the production function from its rate of change \( \frac{dP}{dx} = 100 - 12\sqrt{x} \).

By integrating \( 100 - 12\sqrt{x} \), we get the production increase function \( P(x) = 100x - 8x^{3/2} + C \), where \( C \) is a constant of integration. Determining the indefinite integral allows the firm to establish a relationship that gives insights into how workforce changes alter production over a period or given number of additional employees. This understanding is indispensable for planning and evaluating the outcomes of hiring strategies.

Definite Integral

Definite integrals give the net change of a function over a specific interval. They help in calculating the total increase or decrease in a quantity. In this problem, we calculated the definite integral over the interval from 0 to 25 workers to find the total increase in production. The definite integral of the function \( \int_{0}^{25} (100 - 12\sqrt{x}) \, dx \) gave the total output increase from adding 25 workers.

Within this calculation, the bounds \( 0 \) and \( 25 \) indicate the starting and ending points of the additional workers. Evaluating the definite integral provides essential data, showing how an increase in workforce can tangibly affect production numbers. Companies can then tailor strategies to maximize efficiency based on how output changes within specific parameters.