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Question 1194244: A record company makes an initial investment of $5,000 to prepare songs for a
Musical Album. The cost of manufacturing and recording each disc is 4 dollars. Furthermore, the discography
must pay the singer $1 per record for copyright. It is decided that the selling price
of the disc is 15 dollars.
a. Determine the profit function of the company based on the records sold.
b. Find the balance point.
c. Determine the number of records that must be sold for the company to make a profit.
of $100,000.
d. What are the profits if only 200 discs are sold?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part A
x = number of records made and sold
The initial investment is $5,000.
It costs $4 to make each disc, and there's an extra $1 in cost for copyright purposes. So it really costs 4+1 = 5 dollars per disc.
If the company creates x records, then they spend 5x dollars on top of the $5000 invested.
The cost function is C(x) = 5x+5000
The revenue function is R(x) = 15x because the selling price is $15 per record.
This is the amount of money pulled in before costs are considered.
Profit = Revenue - Cost
P(x) = R(x) - C(x)
P(x) = 15x - (5x+5000)
P(x) = 15x - 5x-5000
P(x) = 10x - 5000
Answer: P(x) = 10x - 5000
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Part B
I'm assuming you're looking for the break-even point.
This occurs when the profit is 0 dollars (see method 1 below)
In other words, it happens when revenue and cost are the same amount (see method 2).
Method 1
Profit = 0
P(x) = 0
10x - 5000 = 0
10x = 5000
x = 5000/10
x = 500
Method 2
Revenue = Cost
R(x) = C(x)
15x = 5x+5000
15x-5x = 5000
10x = 5000
x = 5000/10
x = 500
If 500 records are made and sold, then the company will break even.
They will neither lose money nor gain money.
Answer: 500 records
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Part C
Replace P with 100,000 and solve for x.
P(x) = 10x - 5000
100,000 = 10x - 5000
10x-5000 = 100,000
10x = 100,000 + 5000
10x = 105,000
x = 105,000/10
x = 10,500
Answer: 10,500 records
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Part D
Plug in x = 200 into the profit function
P(x) = 10x - 5000
P(200) = 10(200) - 5000
P(200) = 2000 - 5000
P(200) = -3000
A negative profit means the company lost money.
Notice that x = 200 is smaller than the break-even point we found back in part B.
The company must make more than 500 records to have a positive profit.
Answer: -3000 dollars
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