A firm manufacturing 2000 items.It estimated that the rate of change of production P with the respect to additional number of workers x is given by dP/dx=100-12x^.5.If the firm employees 25 more workers,then the new level of production of the items is?

Question

Elijah Abram · Accepted Answer

New level of production is #3000# As the rate of change of production #P# with the respect to additional number of workers #x# is given by #(dP)/(dx)=100-12x^(0.5)# As #x# changes by #25# i.e. #Deltax=25#, #DeltaP=Deltax xx (dP)/(dx)=25(100-12*25^(0.5))# = #25*(100-12*5)=25*40=1000# and new level of production is #2000+1000=3000#

Ezra Adams · Answer

undefined To find the new level of production of the items when the firm employs 25 more workers, we'll first integrate the given rate of change of production with respect to the number of workers $x$ with respect to $x$ to find the production function. Then, we'll use this function to calculate the new level of production when 25 more workers are employed.

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

Integrating both sides with respect to $x$, we get:

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

$$ P(x) = 100x - 24x^\frac{3}{2} + C $$

Now, let's find the value of the constant $C$ using the initial condition: when $x = 0$, $P = 2000$.

$$ 2000 = 100(0) - 24(0)^\frac{3}{2} + C $$
$$ C = 2000 $$

So, the production function is:

$$ P(x) = 100x - 24x^\frac{3}{2} + 2000 $$

Now, if the firm employs 25 more workers, the new level of production $P_{\text{new}}$ is:

$$ P_{\text{new}} = 100(x + 25) - 24(x + 25)^\frac{3}{2} + 2000 $$

$$ P_{\text{new}} = 100x + 2500 - 24(x + 25)^\frac{3}{2} + 2000 $$

$$ P_{\text{new}} = 100x + 4500 - 24(x + 25)^\frac{3}{2} $$

So, the new level of production of the items when the firm employs 25 more workers is $100x + 4500 - 24(x + 25)^\frac{3}{2}$.