Question

The quantity demanded x for a product is inversely proportional to the cube of the price p for
p > 1.
When the price is $10 per unit, the quantity demanded is 125 units. The initial cost is $140 and the cost per unit is $2. What price will yield a maximum profit?

Answer 1

x = k/p^3
125 = k/10^3 = k/1000
1000 x 125 = k
k = 125000

x = 125000/p^3

The profit is given by G = px - (125 + 2x) = p(125000/p^3) - 125 - 2(125000/p^3) = 125000/p^2 - 250000/p^3 - 125
For maximum profit:
dG/dp = 0
dG/dp = -250000/p^3 + 750000/p^4 = 0
750000 - 250000p = 0
p = 750000/250000 = 3

Therefore the price that will yeaild maximum profit is $3.