Question

If a company has total costs

Answer 1

Answer:

The break-even points are:  x = 500, and x = 70

Step-by-step explanation:

The break even points are those x-values for which the Revenue equals the Cost, so there is no money lost.

Since the formulas for both Revenue and Cost are given, we find the breaking  points by setting the functions equal to each other:

$C(x)=R(x)\\35,000 + 45\,x +0.2 \,x^2=\,615\,x-\,0.8\,x^2\\35,000 + 45\,x +0.2 \,x^2-\,615\,x+\,0.8\,x^2=0\\x^2-570\,x+35,000= 0$

where we have gathered all terms on one side of the equation and written the equation in standard form.

We can use the quadratic formula to solve for "x":

$ax^2+bx+c=0$  has solutions given by: $x=\frac{-b+/-\sqrt{b^2-4ac} }{2a}$

In our case,  $a=1,\,\,b=-570,\,\,and\,\,c=35000$

So the solutions are:

$x=\frac{570+/-\sqrt{570^2-4*35000} }{2}\\x=\frac{570+/-\sqrt{184900} }{2}\\x=\frac{570+/-430 }{2}$

This leads to two solutions:

$x_1=\frac{570+430}{2} = 500\\x_2=\frac{570-430}{2} = 70$

These are the break-even points.