A warehouse having a volume of 10,000 ft3 and a floor area of 1000 ft2 is to be built. The cost
of constructing the floor is $6.00/ft2, the cost of the roof is $10.00/ft2, and the cost of the walls
is $20.00/ft2. If W is the width, L the length, and H the height of the building, what should the
dimensions be such that the cost is minimal?
Obviously H = 10, LW=1000, so L = 1000/W
C = 6*LW + 10*LW + 20*2*WH + 20*2*LH
C = 6000 + 10,000 + 400W + 400(1000/W)
C = 16,000 + 400W + 400,000/W
dC/dW = 400 - 400,000/W^2
max/min C is where dC/dW = 0
400W^2 = 400,000
W^2 = 1000
W = 31.62
L = 31.62
H = 10
To find the dimensions that minimize the cost, we need to consider the costs of each component (floor, roof, and walls) and how the dimensions of the building affect these costs.
Let's break down the different costs involved:
1. Cost of the floor:
The floor area is given as 1000 ft², and the cost of constructing the floor is $6.00/ft². Therefore, the cost of the floor can be calculated as:
Cost of floor = Floor area * Cost per ft²
= 1000 ft² * $6.00/ft²
= $6000.
2. Cost of the roof:
The roof area is the same as the floor area, which is 1000 ft². The cost of constructing the roof is $10.00/ft². So, the cost of the roof can be calculated as:
Cost of roof = Roof area * Cost per ft²
= 1000 ft² * $10.00/ft²
= $10000.
3. Cost of the walls:
The area of the walls can be calculated by subtracting the floor area from the total surface area of the building, which is the sum of the areas of the four walls.
Total surface area = 2 * (Length * Height) + 2 * (Width * Height)
= 2 * (L * H) + 2 * (W * H)
= 2LH + 2WH
The cost of constructing the walls is $20.00/ft². So, the cost of the walls can be calculated as:
Cost of walls = Wall area * Cost per ft²
= (2LH + 2WH) * $20.00/ft²
= $40LH + $40WH.
Now, we can calculate the total cost by summing up the costs of the floor, roof, and walls:
Total cost = Cost of floor + Cost of roof + Cost of walls
= $6000 + $10000 + ($40LH + $40WH)
= $40LH + $40WH + $16000.
To minimize the total cost, we need to find the values of L, W, and H that minimize the expression $40LH + $40WH + $16000.
Unfortunately, the problem as stated does not provide further constraints or requirements, such as the relationship between the dimensions or the minimum or maximum allowed values for the dimensions. Hence, we cannot solve for specific dimensions that minimize the cost without additional information or constraints.