© All Rights reserved @ LearnWithDash
Step 1: Understand the Problem
We are given the rate of change of production with respect to the additional number of workers, represented by the derivative:
$ \frac{dP}{dx} = 100 - 12 \sqrt{x} $.
Also, we know the current production (when $x = 0$) is $2000$. We need to find the new level of production when the firm employs $25$ more workers, i.e., when $x = 25$.
Step 2: Set Up the Integral
From $ \frac{dP}{dx} = 100 - 12 \sqrt{x} $, we can write:
$ dP = \bigl( 100 - 12 \sqrt{x} \bigr)\,dx.
$
We integrate both sides with respect to $x$ to find $P(x)$:
$ \int dP = \int \bigl( 100 - 12 \sqrt{x} \bigr)\,dx.
$
Step 3: Perform the Integration
Integrating term by term, we get:
$ P(x) = 100x - 12 \int \sqrt{x}\,dx + C.
$
Recall that $ \int \sqrt{x}\,dx = \int x^{\frac{1}{2}}\,dx = \frac{2}{3} x^{\frac{3}{2}}. $
Thus,
$ P(x) = 100x - 12 \times \frac{2}{3} x^{\frac{3}{2}} + C
= 100x - 8 x^{\frac{3}{2}} + C.
$
Step 4: Use the Initial Condition
We know that when $x = 0$, $P(0) = 2000$. Substituting $x = 0$ into the expression for $P(x)$ gives:
$ P(0) = 100(0) - 8 \cdot 0^{\frac{3}{2}} + C = C.
$
Hence, $ C = 2000 $.
Therefore, the production function is:
$ P(x) = 100x - 8x^{\frac{3}{2}} + 2000.
$
Step 5: Evaluate at $x = 25$
When $x = 25$, substitute $x = 25$ into $P(x)$ to find the new production level:
$ P(25) = 100 \times 25 - 8 \times (25)^{\frac{3}{2}} + 2000.
$
$ (25)^{\frac{3}{2}} = 25 \times \sqrt{25} = 25 \times 5 = 125.
$
So,
$ P(25) = 2500 - 8 \times 125 + 2000 = 2500 - 1000 + 2000 = 3500.
$
Step 6: Conclusion
The firm’s new level of production, after employing 25 more workers, is $3500$ items.
