You have invested $30,000 and started a business writing greeting cards. Supplies cost $0.02 per card and you are selling each card for $0.50. (Let c represent the number of cards produced and sold.)
a. Write the cost function, c
b. Write the revenue function, R
c. Determine the break-even point. (the entire ordered pair)

- anonymous

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- shadowfiend

If supplies cost $0.02 per card and you make 5 cards, how much does that cost you?

- anonymous

$5.00

- anonymous

I need help on this one

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- shadowfiend

Sorry for the break. How did you get $5 cost to make 5 cards?

- anonymous

.02x5x0.50

- anonymous

or would it be 2.50

- shadowfiend

Ok, sorry. I was talking about just how much it costs, ignoring how much you make off of it. So if we ignore how much you sell the card for, how do you find how much it cost you to make 5 cards?

- anonymous

.02x5=10

- shadowfiend

Right. So for a given number of cards c, how much does it cost just to make that number of cards?

- anonymous

.10x5=.50

- anonymous

so the cost function would be 600 cards produced and sold

- shadowfiend

Er... Wait. No. If you calculated the cost of making 5 cards as 0.02x5, then the cost of making c cards is 0.02c, right?

- shadowfiend

Now, the cost function also takes into account starting costs and then profit made. So, if you sell 5 cards (ignore how much it costs to actually make them), how much do you get for them?

- anonymous

yes but c has to represent the number of cards produced and sold

- shadowfiend

Right, we're dealing with the sold part right now. Call it `n' instead if that makes more sense to you.

- anonymous

$1.00

- anonymous

to make 5 cards

- shadowfiend

So if it costs 0.02n to make n cards, and you get 0.5m when you sell m cards, if you make and cell c cards, how much do you get in total?

- anonymous

$50

- shadowfiend

Well we're talking about some number c of cards, so it can't be an absolute number; it has to be in terms of c.

- anonymous

5x.50=2.50

- shadowfiend

Yes, but for some arbitrary number of cards, c, how much would we make total, including how much it cost us to make the cards and how much we made selling them?

- anonymous

it is costing .02 per card to make and selling them for .50 each. $1.00 to make 5 cards and if we sell 5 cards at .50 then the total would be 3.50

- shadowfiend

Still not clear how you're getting that it costs $1 to make 5 cards. 5 x 0.02 is not $1, it's $0.10.

- anonymous

oh putting my decimal in the wrong place.
then it would be $5.00 to make 5 cards

- shadowfiend

So it costs you $0.10 to make the five cards, you get $3.50 for selling them, so the net amount that you make is $3.50 - $0.10 = $3.40.
You can express this by writing an equation for your net income, N:
N = 0.5c - 0.02c
Then, you have to subtract your initial costs so that you can find the point at which you break even:
N = 0.5c - 0.02c - 30000
Then the point where you break even is where N = 0, so you can solve for c, the number of cards where you have made the same amount as you have spent.

- anonymous

I am selling each card for .50

- anonymous

c(x)=30,000+.02x
r(x)=.50x
.50x=30,000+.02x
-.02= -.02x
.48x=30,000
.48/.48=30,000/.48= 625 cards

- anonymous

is this right???

- anonymous

then the break even point is
.50(625)=(625, 312.50)

- anonymous

is this right???

- shadowfiend

Close -- you messed up your division. 30000/0.48 is 62500. The rest is perfect!

- anonymous

so my break even point in (62500, 31250)

- shadowfiend

Sounds about right.

- anonymous

or should it be (625.00,312.50)

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