Revenue and Cost (Refer to Example 1.) The cost to produce one compact disc is $1.50 plus...

Revenue and Cost (Refer to Example 1.) The cost to produce one compact disc is $1.50 plus a one-time fixed cost of $2000. The revenue received from selling one compact disc is $12.

(a) Write a formula that gives the cost C of producing x compact discs. Be sure to include the fixed cost.

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(b) Write a formula that gives the revenue R from selling x compact discs.

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(c) Profit equals revenue minus cost. Write a formula that calculates the profit P from selling x compact discs.

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(d) How many compact discs need to be sold to yield a positive profit?

EXAMPLE 1 Calculating revenue, cost, and profit

For a computer company, the cost to produce one laptop computer (variable cost) is $1320 plus a one-time cost (fixed cost) of $200,000 for research and development. The revenue received from selling one laptop computer is $1850.

(a) Write a formula that gives the cost C of producing x laptop computers.

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(b) Write a formula that gives the revenue R from selling x laptop computers.

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(c) Profit equals revenue minus cost. Write a formula that calculates the profit P from selling x laptop computers.

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(d) How many computers need to be sold to yield a positive profit?

Solution

(a) The cost of producing the first laptop is

 1320 × 1 + 200,000 = $201,320.

The cost of producing two laptops is

 1320 × 2 + 200,000 = $202,640.

And, in general, the cost of producing x laptops is

 1320 × x + 200,000 = $1320x + 200,000.

Thus C = 1320x + 200,000.

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(b) Because the company receives $1850 for each laptop, the revenue for x laptops is given by R = 1850x.

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(c) Profit equals revenue minus cost, so

 

Thus P = 530x − 200,000.

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(d) To determine how many laptops need to be sold to yield a positive profit, we must solve the inequality P > 0.

 

Because , the company must sell at least 378 laptops. Note that the company cannot sell a fraction of a laptop.