SOLUTION: It cost a company $3500 to produce 3000 pencils whereas it cost $1500 a month even if they don't produce any pencils. If they sell the pencils for $0.75 and the cost, revenue and p

Question 1147464: It cost a company $3500 to produce 3000 pencils whereas it cost $1500 a month even if they don't produce any pencils. If they sell the pencils for $0.75 and the cost, revenue and profit functions are linear, determine how many pencils they must make in a month to break even.
I have gotten R(x)= 0.75x, but I am confused on how to the C(x). I believe that the C(x)=mx + b, but just don't know how to quite get there.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
r(x) = .75 * x
r(x) is the revenue.
x is he number of pencils sold.

you are given that it costs the company 3500 to produce 3000 pencils and that it costs the company 1500 a month even if they don't produce any pencils.

your cost equation is c(x) = 1500 + p * x
c(x) is the cost.
p is price per pencil
x is the number of pencils.

you can see that, when x = 0, the cost is 1500.
that's a flat expense that they have to pay every month in addition to the cost of producing any pencils.

you are given that the cost is 3500 when 3000 pencils are produced.
c(x) = 1500 + p * x becomes 3500 = 1500 + p * 3000
subtract 1500 from both sides of this equation to get:
2000 = p * 3000
divide both sides of this equation by 3000 and solve for p to get:
p = 2/3 of a dollar each = $.666666666..... each.

to break even, r(x) must be equal to c(x).
you get:
.75 * x = 1500 + 2/3 * x
i'll work in fractions, so the formula becomes:
3/4 * x = 1500 + 2/3 * x
subtract 2/3 * x from both sides of this equation to get:
3/4 * x - 2/3 * x = 1500
place all fractions under a common denominator to get:
9/12 * x - 8/12 * x = 1500
combine like terms to get:
1/12 * x = 1500
solve for x to get:
x = 1500 * 12/1 = 18000

the company will break even when 18000 pencils are sold.
r(x) = .75 * 18000 = 13500
c(x) = 1500 + 2/3 * 18000 = 13500
revenue = cost = break even point.

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