Final Answer:
The price that will give the greatest profit is p = $32.
Step-by-step explanation:
Here's how to solve the problem using Lagrange multipliers:
1. Define the optimization problem:
We want to maximize the profit (q), which is the difference between revenue (price * quantity) and cost (quantity * cost per copy).
Subject to the constraint that the quantity (q) depends on the price (p) based on the given equation.
2. Set up the Lagrange function:
L(p, λ) = q(p) - λ(416,000 - 13,000p)
3. Differentiate the Lagrange function with respect to p and λ:
∂L/∂p = -13,000 + 13,000q'(p) - λ * 13,000 = 0
∂L/∂λ = 416,000 - 13,000p - 4q(p) = 0
4. Solve the system of equations:
From the first equation, we can express λ as λ = (13,000 * q'(p)) / 13,000 = q'(p).
Substitute this expression for λ in the second equation:
416,000 - 13,000p - 4q(p) = 0
416,000 - 13,000p - 4 * (416,000 - 13,000p) = 0
3 * 13,000p = 1,664,000
p = 32
5. Check the optimality:
The second derivative of the Lagrange function with respect to p is negative (∂²L/∂p² < 0), confirming that p = 32 is a maximum profit point.
Therefore, the price that will give the greatest profit is p = $32.