What is the break-even point?

Question

This question has 3 parts to it, here is the information for the questions:

The inventor of a new product believes that the cost of producing the product is given by the function C(x) = 1.75x + 7000. If the inventor charges $4 per unit, then her profit for producing and selling x units is given by the function P(x) = 2.25x – 7000

a) What is her profit if she sells 2000 units?
I already did this one and got -$2,500

b) What is her profit if she sells 5000 units?
I did this one as well and got $4,250

c) What is the break-even point for sales stated in the number of units?
This is the one I'm having troubles with. What is a break-even point.. and how do I find it in this scenario?

Thanks in advance.

Answer 1

For Profit use :
P(x) = 2∙25x - 7000
(a)
P(x) = 2∙25x - 7000
P(x) = 2∙25(2000) - 7000
P(x) = 4500 - 7000
P(x) = - $2500.
No profit made when the Women sold 2000 items. She was at a lost to $2500.

(b)
P(x) = 2∙25x - 7000
P(x) = 2∙25(5000) - 7000
P(x) = 11250 - 7000
P(x) = $4,250
Here the profit was $4,250 when 5000 items was sold.

(c)
P(x) = 2∙25x - 7000
Cut even point is then profit and cost equals zero.
P(x) = 2∙25x - 7000
0 = 2∙25x - 7000
7000 = 2∙25x
x = 7000/ 2∙25
x = 3111∙111111.....
It's most likely that product x will be sold as a full unit.
You need to sell 3111∙11111..... to cut even. So selling 3111 products you will be at a slight loss and selling 3112 will put you at a little profit.

Answer 2

The breakeven point in economics is the point at which cost or expenses and income are equal - there is no net loss or gain, one has "broken even".

In this case, set profit = 0. So if you take the profit equation and set p(x) to 0, then x = 7000/2.25 or 3111 units. Sell more than that, make money. Sell fewer, lose money.

Answer 3

The break-even point is the value of x when profit is exactly 0. Sell 1 unit more and you make a profit. Sell 1 unit less and you sustain a loss. So,
P(x) = 0 = 2.25x - 7,000 for break-even
2.25x = 7,000
x = 3,111