Break even point

Question

Suppose the cost of producing x units of a product is given by C(x) = 0.1x 2 + 1600 + 18x and the revenue for the sale of x units is given by R(x) = 100x. Determine the smallest number of units that must be produced and sold to break even.

Answer 1

Hi Nicole,

In this problem we are given:

1. Cost to produce x units C(x) = 0.1x² + 1600 + 18x
2. Revenue for sale of x units R(x) = 100 x

Find: the smallest number of units that must be produced and sold to break even.

To "Break Even" the cost for producing x items would equal the revenue for selling x items.

You are looking for the number of items x that would make the above 2 equations equal to each other.

So what value of x makes them equal

C(x) = R(x)

0.1x² + 1600 + 18x = 100 x

This is a quadratic equation. The zeros of the equation are the possible solutions to the problem.

10(0.1x² - 82x + 1600 = 0)

x²- 820x+16000 = 0

(x-20)(x-800) = 0

x= 20 or 800

Check to see if these values make the cost and revenue the same.

C(x) = 0.1x² + 1600 + 18x

C(20) = 0.1(20)²+1600 +18(20)

C(20) = 2000

R(x) = 100 x

R(20) = 100(20)

R(20) = 2000

When you have made 20 items, you break even.

Now check the other number 800 to see if that also works

C(x) = 0.1x² + 1600 + 18x

C(800) = (0.1)800² + 1600 + 18(800)

C(800) = 80,000

R(x) = 100x

R(800) = 100(800)

R(800)= 80,000

So this number also works.

If you graph the 1st equation, it's a parabola

The 2nd is a straight line.

The intersection of the 2 equations is the answer to your problem.

They intersect twice. Once at 20 and then at 800.

In between 20 and 800, the Revenue is greater than the cost. Before 20 and after 800, Cost exceeds Revenue.

So the first time you break even is the first number units = 20

Answer 2

C(x) = .1x^2 +18x +1600

R(x) = 100x

break even point is when R(x)=C(x)

.1x^2 +18x +1600= 100x

.1x^2 - 82x +1600 = 0

x^2 -820x +16,000= 0

complete the square or use the quadratic formula if you can't see the factors

the following is completing the square

x^2 -820x + 410^2 = -16,000+ 410^2

(x-410)^2 = -16000 + 410^2

x-410 = +/-sqr(410^2 -16,000)

x -410 =+/-sqr( -16,000 + 168,100) = +/-sqr(152100

x = 410+/-390 = 20 or 800

x= 20 = smallest output level to break even

or use a graphing calculator

check the answer

R(20) = 2000

C(20) = .1(20)^2 +18(20)+1600 = 40+360+1600= 2000

it works, as R(x)=C(x) to break even