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The cost of producing x items per day is given by y = 12 + 3x + x^2.
Each item can be sold for $10. The income made when x items are sold is 10*x and the cost of production is 12 + 3x + x^2.
The revenue function is the amount that one gets by selling the items. This is f(x) = 10x
The profit function is the total revenue - total costs, given by P(x) = 10x - 12 - 3x - x^2
At break-even point P(x) = 0
=> 10x - 12 - 3x - x^2 = 0
=> x^2 - 7x + 12 = 0
=> x^2 - 4x - 3x + 12 = 0
=> x(x - 4) - 3(x - 4) = 0
=> (x - 3)(x - 4) = 0
x = 3 and x = 4
We find that if either 3 or 4 items are manufactured the profit is zero. For all other values of x the profit is negative. There is no profit made by producing and selling this item at the given terms (i.e. there is no break-even point).
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