The cost function of a manufacturing concern is linear as follows: C(x) = 25x +

Question

The cost function of a manufacturing concern is linear as follows: C(x) = 25x + 15,000 where C(x) is the total cost in dollars of producing x units, x is the number of units produced .Its revenue function is non-linear, however, as follows: R(x) = -0.01x2 + 50x R(x) is the total revenue in dollars from the sale of x units x is the number of units sold a) What is the profit function? b) Find the breakeven points for this firm. c) What would be the maximum profit, and how many units of their product must be sold to achieve this maximum? The cost function of a manufacturing concern is linear as follows: C(x) = 25x + 15,000 where C(x) is the total cost in dollars of producing x units, x is the number of units produced .Its revenue function is non-linear, however, as follows: R(x) = -0.01x2 + 50x R(x) is the total revenue in dollars from the sale of x units x is the number of units sold a) What is the profit function? b) Find the breakeven points for this firm. c) What would be the maximum profit, and how many units of their product must be sold to achieve this maximum?

Explanation / Answer

Revenue R(x)= -0.01x^2+50x Total cost C(x)= 25x+15000 (a) The profit function = Revenue function - Cost function Profit P(x)= -0.01x^2+25x-15000 [R(x)-C(x)] (b) Breakeven point is the quantum of sales at which total cost is equal to total revenue R(x) = C(x) -0.01x^2+50x = 25x+15000 multiply '-100' both sides x^2-5000x = -2500x-1500000 x^2-2500x+1500000 = 0 x^2-1000x-1500x+1500000 = 0 x(x-1000)-1500(x-1000) = 0 (x-1000)(x-1500) = 0 x = 1000 or 1500 Hence the breakeven point would be at 1000 units and 1500 units (c) Number of units of their product must be sold to achieve this maximum profit P(x) = -0.01x^2+25x-15000 The first derivative of the profit equation becomes -0.02x+25 Set the equation equal to zero as -0.02x+25 = 0 x = -25/-0.02 x = 1250 The maximum profit using the number of units produced calculated above i.e., 1250 P(x) = -0.01x^2+25x-15000 P(x) = -0.01*1250*1250+25*1250-15000 P(x) = 625

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