Question

A present, a firm is manufacturing $1000$ items, It is estimated that the rate of change of production $P$ with respect to the 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 the item is

• A. $2500$
• B. $3000$
• C. $3500$
• D. $4500$

Answer 1

The correct option is A

$2500$

Explanation for the correct option:

Step 1: Integrate the given function:

Given: $\frac{dp}{dx}=100-12\sqrt{x}$

Multiply both sides with $dx$ and integrate both sides.

$$\begin{gathered} \int dp=\int \left(100-12\sqrt{x}\right)dx \\ p=100x-8x^{\frac{3}{2}}+c \end{gathered}$$

Step 2: Find the value of the constant:

Put $x=0$ and $p=1000$ to get the value of $C$.

$$\begin{gathered} 1000=100·0-8·0^{\frac{3}{2}}+c \\ c=1000 \end{gathered}$$

So, $p=100x-8x^{\frac{3}{2}}+1000$

Step 3: Find the new level by putting the value on the increase in workers:

Put $x=25$ in the obtained equation.

$\begin{array}{rcl}p& =& 100·25-8·{25}^{\frac{3}{2}}+1000\\ & =& 2500-1000+1000\\ & =& 2500\end{array}$

Hence, option (A) is the correct answer.