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Question
If the marginal cost (MC) of production of the company is directly proportional to the number of units (x) produced, then find the total cost function, when the fixed cost is ₹ 5,000 and the cost of producing 50 units is ₹ 5,625
Solution
M.C αx
M.C = λx
fixed cost k = ₹ 5000
Cost function C = `int ("M.C") "d"x`
= `int lambdax "d"x`
C = `(lambdax^2)/2 + "k"`
⇒ C = `lambda (x^2/2) + 5000` ........(1)
When x = 50 then C = 5625
5625 = `(lambda(50)^2)/2 + 5000`
5625 – 5000 = `(lambda(2500))/2 = 1250 lambda`
`1250 lambda = 625`
⇒ `lambda = 625/1250 = 1/2`
∴ Required total cost function from equation (1)
C = `1/2(x^2/2) + 5000`
∴ C = `x^2/4 + 5000`
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