# The price of a product varies with the quantity bought as P(x) = 360 - 12*x^2 + 3x. How many products should be bought so that the price per unit is the least?

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### 1 Answer

From the information provided in the problem, the price of x units of the product is given by P(x) = 360 - 12x^2 + 3x. The price unit for x products bought is Pp(x) = 360/x - 12x + 3. The number of units to be bought to minimize the price per unit could be calculate by solving Pp'(x) = 0 for x but it should be kept in mind that the price cannot be negative. The graph of Pp(x) is shown below.

Solving 360/x - 12x + 3 = 0

=> 360 - 12x^2 + 3x = 0

The quadratic equation has a positive root of `(sqrt(1921)+1)/8 ~~ 5.6` , this can be rounded to 6.

**To minimize the price per unit one would have to buy 6 units of the product.**