Optimization Max/Min

Question

Worldwide annual sales of a device in 2012–2013 were approximately q = −6p + 3,060

million units at a selling price of $p per unit. Assuming a manufacturing cost of $30 per unit, what selling price would have resulted in the largest annual profit?

What would have been the resulting annual profit?

Answer 1

profit maximizing price = $270

q= -6p + 3060

-6p = q - 3060

p = 510 - q/6

Revenue = pq = 510q - (1/6)q^2

Marginal Revenue =MR = R' = 510 - q/3

C=30q

Marginal Cost =MC = C' = 30

Max profit is when MR=MC

510 - q/3 = 30

-q/3 = 30-510 = 480

q = 3(480) = 1440 = profit maximizing output level

p=510 - 1440/6 = 510 - 240 = 270 = profit maximizing price

UNLESS the cost is really 30 per unit, which means 30,000,000 per 1 million units, given that 3060 was in units of one million dollars, but there must be a typo

then 510-q/3 = 30,000,000

-q/3 = 29,999,490

q = a negative output level, to maximize profit, profit max level is zero output with zero profits, which avoids losses which come with any positive out put level. Odds are the question didn't intend this problem though.