Answer:
$3240 if they choose to make Box 1
Step-by-step explanation:
First of all, we have to keep in mind that the cost is in square foot, while the dimensions of the boxes are in inches, meaning we have to convert from one length measurement to the other [tex]1 foot = 12 inches[/tex]
Now, the first step is to convert each dimension of the boxes from inches to feet.
For Box 1 :
[tex]Length = 20 in = \frac{20}{12} ft = \frac{5}{3} ft[/tex]
[tex]Width = 6 in = \frac{6}{12} ft = \frac{1}{6} ft[/tex]
[tex]height = 4 in = \frac{4}{12} ft = \frac{1}{3} ft[/tex]
For box 2:
[tex]Length = 15 in = \frac{15}{12} ft = \frac{5}{4} ft[/tex]
[tex]Width = 4 in = \frac{4}{12} ft = \frac{1}{3} ft[/tex]
[tex]Height = 8 in = \frac{8}{12} ft = \frac{4}{3} ft[/tex]
The step two is to calculate the surface area of each box to find out how much material is used to make them. It is important to have in mind the each box has 6 faces being the pairs against each other equals.
then to calculate each of the face areas we follow:
bottom and top : [tex]face area = Length * Width[/tex]
Front and back: [tex]face area = Length * Height[/tex]
Laterals: [tex]face area = Width*Height[/tex]
Box 1:
bottom and top: [tex]face area = \frac{5}{3} * \frac{1}{6} =\frac{5}{18}ft^{2}[/tex]
Front and back: [tex]face area = \frac{5}{3} * \frac{1}{3} =\frac{5}{9}ft^{2}[/tex]
Laterals: [tex]face area = \frac{1}{6} * \frac{1}{3} =\frac{1}{18}ft^{2}[/tex]
Box 2:
bottom and top: [tex]face area = \frac{5}{4} * \frac{1}{3} =\frac{5}{12}ft^{2}[/tex]
Front and back: [tex]face area = \frac{5}{4} * \frac{4}{3} =\frac{5}{3}ft^{2}[/tex]
Laterals: [tex]face area = \frac{1}{3} * \frac{4}{3} =\frac{4}{9}ft^{2}[/tex]
Finally to calculate the total surface area of each box we need to find the sum of the areas of the 6 faces:
Box 1:
[tex]Surface area=\frac{5}{18} *2+\frac{5}{9} *2+\frac{1}{18}*2=\frac{32}{18} =\frac{16}{9}ft^{2}[/tex]
Box 2:
[tex]Surface area=\frac{5}{12} *2+\frac{5}{3} *2+\frac{4}{9}*2=\frac{182}{36} =\frac{91}{18}ft^{2}[/tex]
To calculate the costs per box a simple multiplication is required
Box 1:
$=[tex]\frac{16}{9}*1.10= 1.96[/tex]
Box 2:
$=[tex]\frac{91}{18}*1.10= 5.56[/tex]
Since the company is producing 900 boxes:
Total costs Box 1 : [tex]900*1.96=1764[/tex]
Total costs Box 2: [tex]900*5.56=5004[/tex]
The company would save on materials: [tex]5004-1764=3240[/tex]
$3240 if they choose to make 900 boxes of Box 1.
