# 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?

New level of production is

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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}).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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