QuestionsCategory: QuestionsCase Based question of Application of Derivatives
Harit Thakuri asked 6 months ago

Metal Containers Inc. is reviewing the way it submits bids on U.S. Army contracts. The Army often requests open-top boxes, with square bases and of specified volumes. The Army also specifies the materials for the boxes, and the base is usually made of a different material than the sides. The box is assembled by riveting a bracket at each of the eight corners. For Metal Containers, the total cost of producing a box is the sum of the cost of the materials for the box and the labor costs associated with affixing each bracket.

Instead of estimating each job separately, the company wants to develop an overall approach that will allow it to cost out proposals more easily. To accomplish this, company managers need you to devise a formula for the total cost of producing each box and determine the dimensions that allow a box of specified volume to be produced at minimum cost. Use the following notation to help you solve this problem.

Cost of the material for the base = A per square unit

Cost of the material for the sides = B per square unit

Cost of each bracket = C

Cost to affix each bracket = D

Length of the sides of the base = x

Height of the box = h

Volume specified by the army = V

  1. Write an expression for the company total cost in terms of these quantities.
  2. At the time an order is received for boxes of a specified volume, the cost of the materials and labor will be fixed and only the dimensions will vary. Find the formula for each dimensions of the box so that the total cost is a minimum.

    3.The army requests bids on boxes of 48 cubic feet with base materials costing the container company $12 per square foot and side material costing $8 per square foot. Each bracket costs $5, and the associated labor cost is $1 per bracket. Use your formula to find the dimensions of the box that meet the army's requirement at a minimum cost. What is this cost?

1 Answers
Harit Thakuri answered 6 months ago

Let's denote the following variables:

A: Cost of material for the base per square unit (in dollars per square unit)

B: Cost of material for the sides per square unit (in dollars per square unit)

C: Cost of each bracket (in dollars)

D: Cost to affix each bracket (in dollars)

x: Length of the sides of the base (in units)

h: Height of the box (in units)

V: Volume specified by the Army (in cubic units)

Now, let's write an expression for the total cost of producing a box:

Cost of material for the base: The area of the base is x x, and the cost is A per square unit. So, the cost for the base material is A x^2.

Cost of material for the sides: The area of each side is 4 x h (there are four sides), and the cost is B per square unit. So, the cost for the side material is 4 B x * h.

Cost of brackets: There are 8 brackets, and each costs C. So, the cost for the brackets is 8 * C.

Cost to affix brackets: There are 8 brackets, and each costs D to affix. So, the cost to affix brackets is 8 * D.

Now, the total cost (TC) can be expressed as the sum of these costs: TC=A⋅x^2 + 4⋅B⋅x⋅h + 8⋅C + 8⋅D

Now, Metal Containers wants to minimize the total cost while producing a box with a specified volume V. The volume of the box is given by:

V=x^2⋅h

To minimize the cost with the volume constraint, you can use the volume formula to express one of the variables in terms of the other and substitute it into the cost function. In this case, you can express

h in terms of x using the volume equation: h = V/x^2

Now, substitute this expression for h into the total cost equation:

TC=A⋅x^2 +4⋅B⋅x⋅( V/x^2 )+8⋅C+8⋅D

Given values:

A = $12 per square foot (base material cost)

B = $8 per square foot (side material cost)

C = $5 per bracket

D = $1 per bracket (labor cost per bracket)

V=48 cubic feet (specified volume)

We know that

V=x^2 ⋅h, and the specified volume is 48 cubic feet. To find the dimensions that minimize the cost, we need to express

h in terms of x:

V=x^2 ⋅h

48=x^2⋅h

Solving for h:

h= V/x^2

h = 48/x^2

Now, substitute this expression for h into the total cost equation:

TC=12⋅x^2 +4⋅8⋅x⋅(48/x^2) +8⋅5+8⋅1

Simplify the equation:

TC=12x^2 +192/x+40+8

Now, we want to minimize this cost. Take the derivative of TC with respect to x and set it equal to zero:

dTC/dx=0

=24x− 192/x^2=0

Multiply through by x^2,

24x^3 −192=0

Solving for x:

x=2

Now that we have the value of x, substitute it back into the expression for h:

h = 12

So, the dimensions that minimize the cost are x=2 feet and h=12 feet.

Now, substitute these values into the total cost equation to find the minimum cost:

TC = 48+192+40+8

TC = 228

Therefore, the minimum cost is $288, and the dimensions of the box that meet the army's requirements at this minimum cost are

x=2 feet and h=12 feet.