The weekly demand for a particular commodity is related by p

The weekly demand for a particular commodity is related by p = cubicroot f(x), where p is the price, in hundreds of dollars per unit, when all x units of the commodity are sold. Find the marginal revenue at a weekly sales level of 50 units if f(50) = 64 and f\'(50) = 27.

Solution

Given: p = (f(x))^(1/3) where x denotes the number of commodities sold and p is the price in hundred dollars per unit.; f(50) = 64, f\'(50) = 27

Marginal revenue is the instantaneous rate of change of revenue relative to production at a given production level.

Marginal revenue at a weekly sales level of 50 units = Selling price of 51 items minus selling price of 50 items.-.(1)

Now, SP of 50 items = P(50) = (f(50))^(1/3) = 64^(1/3) = 4 multiplied by 100 dollars = 400.

SP of 51 items = P(51) = P(50) + P\'(50)

P\'(x) = d/dx((f(x))^(1/3) = 1/3 .(f(x))^(-2/3).f\'(x)

P\'(50) = 1/3 .(f(50))^(-2/3).f\'(50) = 0.5625 multiplied by 100 dollars = 56.25

SP of 51 items = P(51) = P(50) + P\'(50) = 400+56.25 = 456.25

Now, as per equation -(1) as above

Marginal revenue at a weekly sales level of 50 units = 456.25-400 = 56.25 dollars