For the cost and price functions below, find the number, q, of units that produces maximum profit; the price, p, per unit that produces maximum profit; and c) the maximum profit, P.
C(q)=70+12q; p=68-2q
To find the number of units that produces maximum profit, we need to find the value of q that maximizes the profit function. The profit function, P(q), is given by:
P(q) = p(q)q - C(q)
Substituting p(q) = 68 - 2q and C(q) = 70 + 12q, we get:
P(q) = (68 - 2q)q - (70 + 12q)
Expanding and simplifying, we get:
P(q) = -2q^2 + 56q - 70
To find the maximum point of this function, we can find the vertex by using the formula:
q = -b/2a
where a = -2, b = 56. Substituting these values, we get:
q = -56/(2*(-2)) = 14
So, the number of units that produces maximum profit is 14.
To find the price per unit that produces maximum profit, we can use the price function:
p(q) = 68 - 2q
Substituting q = 14, we get:
p(14) = 68 - 2(14) = 40
So, the price per unit that produces maximum profit is $40.
To find the maximum profit, we can substitute q = 14 and p = 40 into the profit function:
P(14) = (40)(14) - (70 + 12(14)) = $168
So, the maximum profit is $168.