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solmaris [256]
2 years ago
13

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.
Mathematics
1 answer:
mina [271]2 years ago
7 0

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.

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