Question:
At present, a firm manufactures 1099 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by $ \frac{dP}{dx}=100-12\sqrt{x}. $ . If the firm empower 25 more workers, then the new level of production of items is
At present, a firm manufactures 1099 items. It is estimated that the rate of change of production P with respect to additional number of workers x is given by $ \frac{dP}{dx}=100-12\sqrt{x}. $ . If the firm empower 25 more workers, then the new level of production of items is
Updated On: Jun 23, 2024
- $ 2000 $
- $ 2500 $
- $ 3000 $
- $ 3500 $
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The Correct Option is B
Solution and Explanation
Given, $ \frac{dP}{dx}=100-12\sqrt{x} $
$ \Rightarrow $ $ dP=(100-12\sqrt{x})dx $
On integrating both sides, we get
$ P=100x-12\left( \frac{{{x}^{3/2}}}{3/2} \right)+C $
$ \Rightarrow $ $ P=100x-8{{x}^{3/2}}+C $ ..(i)
At initially $ x=0,\,\,\,\,P=1000 $
Then, from E (i) we get $ 1000=100\times 0-0+C $
$ \Rightarrow $ $ C=1000 $
On putting the value of C in E (i), we get
$ P=100x-8{{x}^{3/2}}+1000 $
Now, at $ x=25, $ $ P=100(25)-8{{(25)}^{3/2}}+1000 $
$=2500-8(125)+100 $
$=3500-1000=2500 $
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations