Hello there. To solve this question, we'll need to set up an equation that gives you the profit of the commodity.
First, the profit P(x) is given by the difference between the revenue R(x) and the costs C(x)
P(x) = R(x) - C(x)
R(x) is given by the number of units sold multiplied by the price of the unit, in this case, x * p = (1350 - 45p)p = 1350p - 45p².
The cost C(x) is given by $5 per unit sold and a fixed cost of $2000 per month, it is, 5x + 2000 = 5(1350 - 45p) + 2000 = 8750 - 225p.
Then the profit will be:
P(x) = 1350p - 45p² - (8750 - 225p)
P(x) = 1350p - 45p² - 8750 + 225p
P(x) = -45p² + 1575p - 8750.
To find the maximum value of this function, we'll use the formula for quadratic functions: The number x that maximizes ax² + bx + c is given by -b/2a and max (ax² + bx² + c) = -(b² - 4ac)/4a, for a not equal to zero.
Knowing that a = -45, b = 1575 and c = -8750, we get:
p_max = -1575/(2*(-45))
p_max = -1575/-90
p_max = $17.50
The price p per unit that should be charged to the consumers in order to obtain a maximum monthly profit is $17.50