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As the price of a product changes, the number of units consumers are willing to buy changes. This is called the demand and the number of units sellers and willing to sell is known as the supply. For most products demand increases as the price decreases and supply increases as the price increases. The cost function of a product C(x) is the amount that has to be spent by the producer to make a quantity x of the product.
In the question the demand and cost functions for the software sold are provided. The demand function is Q(p) = 1000*(200 - p). The cost function is given by TC(p) = 5000000 + 40*p.
Let the number of copies of the software sold at price p be represented by x.
x = 1000*(200 - p)
(200 - p) = x/1000
p = 200 - x/1000
The revenue of the company when x copies are sold is R(x) = x*p or
R(x) = x*(200 - x/1000) = 200x - x^2/1000
If the company is selling x copies, the additional revenue for each extra copy sold is equal to R'(x) = 200 - x/500. This is the marginal revenue function.
The marginal cost function of the the software is given by C'(x) = 40
To maximize profits, it should produce x copies, at which point the marginal revenue and marginal cost is equal.
Solving 40 = 200 - x/500 gives,
x/500 = 160
x = 80000
If sales are to equal 80000, the price of the software has to be $120.
The monthly profit in this case is 120*80000 - (5000000 + 40*80000) = 1400000
Quantum Dynamics should charge $120 for its software and the monthly profit it can make is equal to $1400000.
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