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

We are told that the firm is already producing $$2000$$ items. We introduce a variable $$x$$ to denote the extra number of workers hired, and we are given the rate of change of production with respect to this variable:

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

The firm decides to hire $$25$$ more workers, so $$x$$ will vary from $$0$$ to $$25$$. We must find the new production $$P(25)$$ knowing that $$P(0)=2000$$.

To obtain $$P(x)$$ from its derivative, we integrate. We first state the basic rule:

If $$\dfrac{dP}{dx}=f(x)$$, then $$P(x)=\displaystyle\int f(x)\,dx + C,$$ where $$C$$ is the constant of integration determined by an initial condition.

Applying this rule, we write

$$P(x)=\int\left(100-12\sqrt{x}\right)\,dx + C.$$

We integrate term-by-term. For the first term, $$\int 100\,dx=100x.$$ For the second term, we recall that $$\sqrt{x}=x^{1/2}$$ and use the power rule $$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C.$$ Hence

$$\int -12\sqrt{x}\,dx=-12\int x^{1/2}\,dx=-12\cdot\frac{x^{3/2}}{3/2}=-12\cdot\frac{2}{3}x^{3/2}=-8x^{3/2}.$$

Combining the two integrated parts gives

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

We now determine the constant $$C$$ by imposing the initial condition $$P(0)=2000.$$ Substituting $$x=0,$$ we get

$$2000=P(0)=100(0)-8(0)^{3/2}+C\quad\Longrightarrow\quad C=2000.$$

Thus the full expression for production, in terms of the additional workers $$x,$$ is

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

Now we evaluate this function at $$x=25$$ because the firm hires $$25$$ more workers:

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

We simplify step by step. First compute the integer multiplication:

$$100(25)=2500.$$

Next, we handle $$25^{3/2}.$$ Since $$25=5^2,$$ we have

$$\sqrt{25}=5\quad\text{and hence}\quad 25^{3/2}=(25^{1/2})^{3}=5^{3}=125.$$

Using this, we get

$$-8(25)^{3/2}=-8(125)=-1000.$$

Now we combine all the pieces:

$$P(25)=2500-1000+2000.$$

Adding the numbers sequentially, we first compute $$2500-1000=1500,$$ and then

$$1500+2000=3500.$$

Thus, after employing $$25$$ additional workers, the firm’s new level of production is

$$3500\text{ items}.$$

Hence, the correct answer is Option A.