Final Answer:
The price that will give the greatest profit is $20.
Step-by-step explanation:
To find the price that will give the greatest profit, we need to maximize the revenue function R(p) = p*q(p), where q(p) is the demand function.
The demand function is given by q(p) = 416,000 - 13,000p. The cost of producing each copy is $4, so the cost function is C(p) = 4q(p) = 1,664,000 - 52,000p.
Using the method of Lagrange multipliers, we can set up the following equation:
∇R(p) = λ∇C(p)
Differentiating R(p) and C(p) with respect to p, we get:
13,000 - 26,000λp = 0
Solving for p, we get:
p = 1/2λ
Substituting this value of p in the demand function, we get:
q = 416,000 - 13,000(1/2λ) = 208,000 - 6,500λ
The revenue function can now be written as:
R(λ) = p*q = (1/2λ)(208,000 - 6,500λ)
Differentiating R(λ) with respect to λ, we get:
R’(λ) = 104,000 - 13,000λ
Setting R’(λ) = 0, we get:
λ = 8
Substituting this value of λ in p = 1/2λ, we get:
p = $20
Therefore, the price that will give the greatest profit is $20.