Math, asked by rupesh2335be20, 5 months ago

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

Answered by nagagowshika
2

Answer:

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Step-by-step explanation:

hope u help this

Answered by prateekmishra16sl
2

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 = \frac{-90}{-0.01}

Maximum Profit =  9000

q₁  = -1/λ = 100

q₂  = -2/λ = 200

q₃  = -3/λ = 300

#SPJ3

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