A manufacturer can produce three distinct products in quantities q1,q2 and q3, respectively, and there by derive a profit p(q1,q2, q3) = 2q1+8q2+24q3. Find q1, q2, q3
that maximize profit if production is subject to the constraint q12+ 2q22+4q32= 450000.
Answers
Answer:
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Step-by-step explanation:
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Answer: The values of q₁, q₂, q₃ for maximum profit are 100 , 200 and 300 respectively.
Step-by-step explanation:
Profit ⇒ P(q₁,q₂, q₃) = 2q₁+8q₂+24q₃
Constrain ⇒ (q₁)²+ 2(q₂)²+4(q₃)² = 450000
Lagrangian function ⇒ L : 2q₁+8q₂+24q₃ + λ(q₁²+ 2q₂²+4q₃² - 450000)
λ ⇒ Lagrangian Multiplier
dL/dq₁ = 0
⇒ 2 + 2λq₁ = 0
⇒ q₁ = -1/λ
dL/dq₂ = 0
⇒ 8 + 4λq₂ = 0
⇒ q₂ = -2/λ
dL/dq₃ = 0
⇒ 24 + 8λq₃ = 0
⇒ q₃ = -3/λ
dL/dλ = 0
⇒ (q₁)²+ 2(q₂)²+4(q₃)² = 450000
Substitute values of (q₁,q₂, q₃) in terms of λ
⇒ (-1/λ)²+ 2(-2/λ)²+4(-3/λ)² = 450000
⇒ 1/λ²+ 8/λ²+ 36/λ² = 450000
⇒ 45/λ² = 450000
⇒ λ² = 45/450000
⇒ λ² = 0.0001
⇒ λ = ± 0.01
Profit = 2q₁+8q₂+24q₃
Profit = -2/λ - 16/λ - 72/ λ = -90/λ
For maximum profit , λ should be negative.
Therefore, for maximum profit,
⇒ λ = -0.01
Maximum Profit =
Maximum Profit = 9000
q₁ = -1/λ = 100
q₂ = -2/λ = 200
q₃ = -3/λ = 300
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