Suppose you are the marketing manager for the Fruit of the Loom. An individuals' inverse demand for Fruit of the Loom women's underwear is estimated to be P = 25 - 3Q (in cents). If cost to Fruit of the Loom to producing women's underwear is C(Q) = 1 + 4Q (in cents), compute the number of women's underwear that should be packaged together.
A. 1
B. 3
C. 4
D. 7
To determine the optimal number of women's underwear that should be packaged together, we need to find the quantity that maximizes Fruit of the Loom's profit. Profit is calculated by subtracting the cost of production from the revenue earned from selling the product.
The revenue can be calculated using the inverse demand equation: P = 25 - 3Q. Since revenue is equal to price multiplied by quantity, we can rewrite the equation as follows: R = (25 - 3Q)Q = 25Q - 3Q^2.
The cost of production is given by the function C(Q) = 1 + 4Q.
Profit (Π) is equal to revenue minus cost: Π = R - C = (25Q - 3Q^2) - (1 + 4Q) = -3Q^2 + 21Q - 1.
To find the optimal quantity, we need to take the derivative of the profit function with respect to Q and set it equal to zero. This will give us the critical point where the profit is maximized.
dΠ/dQ = -6Q + 21 = 0.
Solving for Q, we have:
-6Q + 21 = 0
-6Q = -21
Q = -21 / -6
Q = 3.5
Since the quantity of women's underwear must be a whole number, we round down to the nearest whole number, which is 3.
Therefore, the optimal number of women's underwear to be packaged together is B. 3.