For the cost and price functions below find a the number q o

For the cost and price functions below, find a) the number, q, of units that produces maximum profit; b) the price, p, per unit that produces maximum profit; and c) the maximum profit, P C(q) = 80 + 12q; p = 56-2q a) The number, q, of units that produces maximum profit is q = b) The price, p, per unit that produces maximum profit is p-S c) The maximum profit is Ps

Solution

q is the number of sales. The price p can be less if sales are high, as given by
p = 56 – 2q
The “takings” received is usually called revenue R and is price * sales.
R = p*q = 56q – 2q^2
But of course the revenue is not the same as the profit, because it will cost something to make and distribute the items for sale.
The cost C might be a fixed amount to buy materials plus an amount for delivering items. C(q) = 80 + 12q
Profit P = Revenue R – Cost C
P(q) = 56q – 2q^2 – (80 + 12q) = 44q – 2q^2 – 80 ............(1)
A graph of this would be an inverted parabola so expect a maximum.
Differentiate and set to zero to find max.
dP/dq = 44 – 4q = 0
The theoretical number of sales of units that produces maximum profit is
q = 11  
p = 56 – 2q = 34
The theoretical maximum profit using q = 11 in (1) would be
P(max) = 162