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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Cost A cargo container (in the shape of a rectangular solid) must have a volume of 480 cubic feet. The bottom will cost $5 per square foot to construct and the sides and the top will cost $3 per square foot to construct. Find the dimensions of the container that has a minimum cost.

To determine

To calculate: The dimensions of the container that has a minimum cost which is having volume as 480cubicfeet, cost of construction per square foot of the bottom $5 and cost of construction per square foot of the side and top $3

Explanation

Given Information:

The volume of the cargo container is 480 cubic feet.

The cost of construction per square foot of the bottom is $5.

The cost of construction per square foot of the sides and top is $3.

Formula used:

The Lagrange Multiplier,

f(x,y,z)=λg(x,y,z)

Where, f(x,y,z)andλg(x,y,z) are the function and the constraint and λ is the Lagrange Multiplier.

Step 1: Write the function and the constraint values.

Step 2: Partially differentiate 'f'and'g' with respect to x,yandz.

Step 3: Using the primary formula, find the value of λ in each case.

Step 4: Find the relation between the dimensions x,yandz.

Step 5: Substitute them half into the constraint equation.

Step 6: Substitute the value of the found dimension in the constraint equation to obtain the other dimension.

Calculation:

The volume of the container which is in the shape of the rectangular solid is given by,

V=xyz

The volume of the cargo container is 480 cubic feet.

Now, the total cost of the container is given by,

Total cost=Cost of the bottom the area of the bottom                 + Cost of the sidesthe area of the sides                 + Cost of the other sides the area of the other sides                 + Cost of the top the area of the top

So,

Total cost=($5)xy+($3)2xz+($3)2yz+($3)xy=8xy+6xz+6yz

Here, given that g(x,y,z) as V=xyz and f(x,y,z) as Total cost=8xy+6xz+6yz.

Now partially differentiating f(x,y,z) with respect to x,yandz,

dfdx=ddx(8xy+6xz+6yz)dfdx=8y+6z

And,

dfdy=ddy(8xy+6xz+6yz)dfdy=8x+6z

And,

dfdz=ddz(8xy+6xz+6yz)dfdz=6x+6y

Now partially differentiating g(x,y,z) with respect to x,yandz

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