Question

The revenue function R(x) and the cost function C(x) for a particular product are given. These functions are valid only for the specified range of values. Find the number of units that must be produced to break even.
R(x)=200x-x^2; C(x)=35x+4550; 0< x < 100. The < symbol has a line under it.
The manufacturer must produce _______ units to break even.

Answer 1

Answer:

The manufacturer must produce 35 units to break even.

Step-by-step explanation:

The break even function is when those functions, the revenue R(x) and the cost C(x), are equal. So

$R(x) = C(x)$

$200x - x^{2} = 35x + 4550$

$x^{2} + 35x - 200x + 4550 = 0$

$x^{2} - 165x + 4550 = 0$

So we have to solve this quadratic function to find the number of units that the manufacturer must produce to break even:

Solving a quadratic function:

Given a second order polynomial expressed by the following equation:

$ax^{2} + bx + c, a\neq0$.

This polynomial has roots $x_{1}, x_{2}$ such that $ax^{2} + bx + c = (x - x_{1})*(x - x_{2})$, given by the following formulas:

$x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}$

$x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}$

$\bigtriangleup = b^{2} - 4a$

In this problem, we have that:

$x^{2} - 165x + 4550 = 0$

So

$a = 1, b = -165, c = 4550$

$\bigtriangleup = (-165)^{2} - 4(1)(4550) = 9025$

$x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a} = \frac{-(-165) + \sqrt{9025}}{2(1)} = 130$

$x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a} = \frac{-(-165) - \sqrt{9025}}{2(1)} = 35$

The manufacturer produces between 0 and 100 units. So the answer is 35.

The manufacturer must produce 35 units to break even.