Your task is to design a rectangular industrial warehouse consisting of three separate spaces of equal size.?
The wall materials cost $58 per linear foot and your company has allocated $27840 for those walls.
1) The dimensions which use all of the budget and maximize total area are
length L=.........
width W=........
2) Each of the three (equal size) compartments has area =........
To design a rectangular warehouse consisting of three separate spaces, we need to find the dimensions that maximize the total area while staying within the allocated budget for wall materials. Let's follow the steps to find the answers to the questions:
1) To find the dimensions that use all of the budget and maximize the total area, we need to determine the perimeter and the corresponding dimensions.
Let's assume the length of each compartment is L, and the width is W.
The total cost for wall materials is given as $27840, and the cost per linear foot is $58.
The perimeter of each compartment is 2L + 2W. Since there are three compartments, the total perimeter is 3*(2L + 2W) = 6L + 6W.
At $58 per linear foot, the total cost for wall materials is equal to the total perimeter multiplied by the cost per linear foot:
$58 * (6L + 6W) = $27840
Now we can solve this equation to find the values of L and W:
6L + 6W = 27840 / 58
6L + 6W = 480
To maximize the total area, we can use calculus and the fact that the maximum area is achieved when the length and width are equal:
L = W
Substituting L for W in the equation above:
6L + 6L = 480
12L = 480
L = 40
Therefore, the length (L) of each compartment is 40 ft.
To find the width (W), we can substitute the value of L back into the equation:
6L + 6W = 480
6(40) + 6W = 480
240 + 6W = 480
6W = 480 - 240
6W = 240
W = 40
Therefore, the width (W) of each compartment is also 40 ft.
So, the dimensions that use all of the budget and maximize the total area are:
Length (L) = 40 ft
Width (W) = 40 ft
2) Each of the three compartments has the same area, which is equal to the length multiplied by the width:
Area = L * W
Area = 40 ft * 40 ft
Area = 1600 square feet
Therefore, each of the three compartments has an area of 1600 square feet.