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 64 units. The initial cost is $140 and the cost per unit is $4. What price will yield a maximum profit? (Round your answer to two decimal places.)

$______

Answer 1

$6.00

Step-by-step explanation:

Given data

quantity demanded ( x ) ∝ 1 / p^3 for p > 1

when p = $10/unit , x = 64

initial cost = $140, cost per unit = $4

Determine the price that will yield a maximum profit

x = k/p^3 ----- ( 1 ). when x = 64 , p = $10 , k = constant

64 = k/10^3

k = 64 * ( 10^3 )

= 64000

back to equation 1

x = 64000 / p^3

∴ p = 40 / ∛x

next calculate the value of revenue generated

Revenue(Rx) = P(price ) * x ( quantity )

= 40 / ∛x * x = 40 x^2/3

next calculate Total cost of product

C(x) = 140 + 4x

Maximum Profit generated = R(x) - C(x) = 0

= 40x^2/3 - 140 + 4x = 0

= 40(2/3) x^(2/3 -1) - 0 - 4 = 0

∴ ∛x = 20/3 ∴ x = (20/3 ) ^3 = 296

profit is maximum at x(quantity demanded ) = 296 units

hence the price that will yield a maximum profit

P = 40 / ∛x

= ( 40 / (20/3) ) = $6