Is my data correct?
For package design I will use a cylinder. The Equation for the surface area of a cylinder is A = 2 π r h + 2 π r² . The equation for the volume of a cylinder is V = π r² h. I have chosen a cylinder for my cereal box because I believe a cylinder can hold the most cereal in the most efficient way.
Dimentions of a regular cereal box
length 19cm
width 6cm
height 27cm
Volume of a rectangular prism equation V = l · w · h.
V = 19 · 6 · 27 = 1539.
The Volume of a regular cereal box is 3078 cubic inches
Surface area of a rectangular prism formula A = 2(w · l + h · l + h · w)
A = 2( 6 · 19 + 27 · 19 + 27 · 6)
A = 1578.
The Surface area is 1578cm²
Conversion factor 1inch = 2.5 cm
I will divide the surface area by 2.5 to get how many inches.
1578 / 2.5 = 631.2
I will then multiply 631.2 by .5 to see how much it would cost to make a regular cereal box.
631.2 · .5 = 315.6
To Manufacture a regular sized cereal box it would take $31.56.
My Cereal box is a cylinder
Length 4cm
Width 20cm
Height 11cm
V=π r² h
V= π ·102 · 11 ≈3455.75192
The volume of my cylinder cereal box is 3455.75 cubic inches
Surface area of a cylinder A = 2 · π · r · h + 2 · π · r²
A = 2 · π · 10 · 11 + 2 · π · 10² ≈ 1319.46891
The Surface area of my cylinder is 1319.47 cm²
I will follow the same steps I did before
Convert from cm to inches
1319.47 / 2.5 = 527.788
Then 527.788 · .5 = 263.894
It would take approximately $26.39 to manufacture my cylinder cereal box.
you are correct that a cylinder has less surface area than a rectangular prism.
To verify if your data is correct, let's go through the calculations step by step.
First, let's calculate the volume of the regular cereal box using the formula V = l · w · h:
V = 19 cm · 6 cm · 27 cm = 3078 cm³
Now, let's calculate the surface area of the regular cereal box using the formula A = 2(w · l + h · l + h · w):
A = 2(6 cm · 19 cm + 27 cm · 19 cm + 27 cm · 6 cm) = 1578 cm²
To convert the surface area from cm² to inches², let's use the conversion factor 1 inch = 2.5 cm:
1. Divide the surface area in cm² by 2.5: 1578 cm² / 2.5 cm = 631.2 inches²
Now, let's calculate the manufacturing cost for the regular cereal box using the formula:
Manufacturing cost = Surface area in inches² × 0.5
Manufacturing cost = 631.2 inches² × 0.5 = 315.6 dollars
The manufacturing cost for the regular cereal box is approximately $31.56.
Let's now move on to the calculations for your cylinder cereal box.
The dimensions of your cylinder cereal box are: length = 4 cm, width = 20 cm, height = 11 cm.
To calculate the volume of the cylinder, we'll use the formula V = π r² h:
V = π · 10² · 11 ≈ 3455.75 cm³
To calculate the surface area of the cylinder, we'll use the formula A = 2 π r h + 2 π r²:
A = 2 · π · 10 · 11 + 2 · π · 10² ≈ 1319.47 cm²
Now, let's convert the surface area from cm² to inches² using the conversion factor:
1319.47 cm² / 2.5 cm = 527.788 inches²
Finally, let's calculate the manufacturing cost for the cylinder cereal box:
Manufacturing cost = Surface area in inches² × 0.5
Manufacturing cost = 527.788 inches² × 0.5 = 263.894 dollars
Therefore, the manufacturing cost for your cylinder cereal box is approximately $26.39.
To check if your data is correct, you can compare the calculated values with your initial assumptions and equations.