At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of productio

Q: Wait, why do we need the constant $C$? Can't we just integrate and get the answer?

Great question! Integration gives you a 'family' of curves, not just one. The constant $C$ is the 'DNA' of your specific factory—it anchors your mathematical model to the real-world reality that you started with 2000 items. Without $C$, you'd only be calculating the *increase* in production, not the *total* production. Always respect the constant!

Q: Why is the derivative $dP/dx$ decreasing as $x$ increases? Does that make sense?

Absolutely! This is a beautiful example of the 'Law of Diminishing Returns.' In a real factory, as you add more workers, they might start getting in each other's way, or you might run out of machines for them to use. The math reflects this reality: the rate of production growth slows down as $x$ gets larger. It's not just abstract algebra; it's economics in action.

Q: Is there a shortcut? Can I use definite integration from $0$ to $25$?

You are thinking like a pro! Yes, you can calculate the change in production as $\Delta P = \int_{0}^{25} (100 - 12\sqrt{x}) dx$. Then, your final production is simply $P_{final} = P_{initial} + \Delta P$. This is a perfectly valid and elegant way to bypass finding $C$ explicitly. Both methods lead to the same destination!

Question

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is

Options:

(A) 2500

(B) 3000

(C) 3500

(D) 4500

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the floor of a bustling manufacturing plant. The firm is currently churning out 2000 items, and we are tasked with modeling the production growth as new workers are added.

The problem provides the rate of change of production $P$ with respect to the number of workers $x$:

$$ \frac{dP}{dx} = 100 - 12\sqrt{x} $$

In engineering terms, this is the 'marginal production.' It represents the instantaneous rate at which production increases for every infinitesimal addition of labor.

### The Integration Journey

To determine the total production $P$, we must accumulate these marginal gains. We begin by separating the variables:

$$ dP = (100 - 12x^{1/2}) dx $$

Next, we integrate both sides of the equation:

$$ \int dP = \int (100 - 12x^{1/2}) dx $$

Applying the power rule for integration, the integral of $100$ is $100x$. For the term $12x^{1/2}$, we increase the exponent to $3/2$ and divide by the new exponent:

$$ P = 100x - 12 \left( \frac{x^{3/2}}{3/2} \right) + C $$

Simplifying the expression, we obtain the general production function:

$$ P = 100x - 8x^{3/2} + C $$

### The Constant of Reality

The constant $C$ represents the 'base production'—the initial output of 2000 items before any new workers are hired. We use the initial condition where $x=0$ and $P=2000$:

$$ 2000 = 100(0) - 8(0)^{3/2} + C $$

This calculation confirms that $C = 2000$. Our specific production function is therefore:

$$ P(x) = 100x - 8x^{3/2} + 2000 $$

### The Final Calculation

To find the production level after hiring 25 new workers, we substitute $x=25$ into our function:

$$ P = 100(25) - 8(25)^{3/2} + 2000 $$

Since $25^{3/2} = (\sqrt{25})^3 = 5^3 = 125$, the equation becomes:

$$ P = 2500 - 8(125) + 2000 $$

Calculating the terms, we find $P = 2500 - 1000 + 2000$. The final production total is:

**$P = 3500$ items**

Select Answer:

Visualized Solution

Initial Production State

The Rate of Change

Separating Variables

Integrating the Rate

Performing Integration

Simplifying the Equation

Finding the Constant $C$

Evaluating $C$

The Specific Production Function

Substituting $x = 25$

Calculating the Exponent

Final Arithmetic

Final Production Level

The Sigma Insight: Variable Separable Method

Analyzing the Setup

The Integration Journey

The Constant of Reality

The Final Calculation

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