Suppose the profit from the sale of x units of a product is

Suppose the profit from the sale of x units of a product is P=6400x-18x^2-400. (What levels of production will yield profit of $274,600?)

Solution

Assuming that the entire production is sold ( since x is sales and we need to know production), the profit will be $ 274600, when P =  274600, i.e. when 6400x - 18x² - 400 =  274600, Then 18x² - 6400x + 274600 + 400 = 0 or, 18x² - 6400x + 27500 = 0 or, 9x² - 3200x + 137500 = 0. Then x = [ 3200 ± {( - 3200)² - (4*9*137500)}] / (2*9) or, x = [ 3200 ± ( 10240000 - 4950000] / 18 = ( 3200 ± 5290000) / 18 = (3200 ± 2300) /18 i.e. either x = 5500/18 = 305.56 (approx) or, x = 900/18 = 50 Since x has to be an integer ( production cannot be in a fraction), we should have sale and production of 50 units for a profit of $ 274600.

NOTE: We have used the fact that the roots of the equation, ax² +bx + c = 0 are [ -b ± (b² -4ac)] / 2a

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