For the next two questions, you will be finding the maximum volume of an open-top box given that the sides cost 5 dollars a square foot to make the bottom costs 40 dollars a square foot to make and you have 480 dollars to spend. Question 1; the Lagrange multiplier set up for this problem is: (note: x,y, and z are the dimensions of the box with the bottom having area xy) The maximum volume given the constraint is where x s F(x,y,λ)=xyz−λ(480−10xz−10yz−40xy) F(x,y,λ)=2xz+2yz+zy−λ(480−xyz) F(x,y,λ)=xyz−λ(480−2xz−2yz−xy) F(x,y,λ)=10xz+10yz+40xy−λ(480−xyz)

Question

07-02-2024
Mathematics

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For the next two questions, you will be finding the maximum volume of an open-top box given that the sides cost 5 dollars a square foot to make the bottom costs 40 dollars a square foot to make and you have 480 dollars to spend. Question 1; the Lagrange multiplier set up for this problem is: (note: x,y, and z are the dimensions of the box with the bottom having area xy) The maximum volume given the constraint is where x s F(x,y,λ)=xyz−λ(480−10xz−10yz−40xy) F(x,y,λ)=2xz+2yz+zy−λ(480−xyz) F(x,y,λ)=xyz−λ(480−2xz−2yz−xy) F(x,y,λ)=10xz+10yz+40xy−λ(480−xyz)

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sqdancefan sqdancefan · Answer 1 · 2024-02-07T03:53:31+01:00

Answer:

(a) F(x,y,z,λ) = xyz−λ(480−10xz−10yz−40xy)

Step-by-step explanation:

You want the Lagrangian function for the problem of maximizing the product xyz given the constraint 10xz +10yz +40xy = 480.

Lagrangian

The Lagrangian function for the problem of maximizing or minimizing f(x) subject to the constraint g(x)=0, is written ...

L(x, λ) = f(x) +λ·g(x) . . . . . . where λ is the "Lagrange multiplier"

Application

Here, the function "f(x)" is a function of x, y, z, and the constraint is that the total cost is $480.

f(x, y, z) = xyz . . . . . . the volume of the box, which we want to maximize

g(x, y, z) = 40xy +2(5)(x+y)z -480 . . . . . . . excess of cost over $480

g(x, y, z) = 10xz +10yz +40xy -480

The Lagrangian function for this problem is then ...

F(x, y, z, λ) = xyz -λ(480 -10xz -10yz -40xy) . . . . matches choice A

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Additional comment

The solution to the four equations ∂F/∂x = 0, ∂F/∂y = 0, ∂F/∂z = 0, ∂F/∂λ = 0 is x = y = 2, z = 8, λ = -1/10. The maximum volume is 32 cubic feet.

As with many problems of this sort, you can solve it mentally. The solution that minimizes cost is the one that makes the costs of opposite sides and the bottom each be equal at 1/3 the total. That means the bottom costs $160, so is 160/40 = 4 square feet. The bottom edge of each side is √4 = 2 ft, so the height is $160/($10/ft²)/(2 ft) = 8 ft.