a soup manufacturer is trying to decide between two containers for packaging soup for sale.
option A- the soup is packaged in a cylindrical can made of steel and tin. The cylinder has a height of 4.25 in. and a radius of 1.625 in. The packaging material costs $0.00043 per square inch
Option B- The soup is packaged in a container shaped like a rectangular prism and is made of paperboard, plastic, and aluminum foil. The container has a length of 3.5 in., a width of 1.5 in., and a height of 6.7 in. The packaging material costs $0.00036 per square inch.
Based on the cost of the packaging, which container is cheaper. Showing work is appreciated
Cylinder:
Total area of top and bottom 2 (pi r^2)
Total area of side = 2 pi r h
Total area = 2 pi r (r+h)
Box:
top and bottom area = 2 ( length * width)
sides = 2 * length*height
ends = 2 * width * height
so total area of box = 2 (Lw +Lh +wh)
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multiply each by cost/in^2 to compare total costs
the area of a cylinder is 2πr(r+h)
A: 2π*1.625(1.625+4.25) * 0.00043 = $0.0258
the area of a prism is 2(lw+wh+lh)
B: 2(3.5*1.5 + 1.5*6.7 + 3.5*6.7) * 0.00036 = $0.0279
PS, no one in their right mind stores soup in a box.
thank you guys for the help
To determine which container is cheaper based on the cost of packaging, we need to calculate the surface area of each container and then multiply it by the cost per square inch of the packaging material.
First, let's calculate the surface area of option A, the cylindrical can:
The formula to calculate the surface area of a cylinder is given by:
A = 2πr(r + h)
where A is the surface area, r is the radius, and h is the height.
Using the given values:
r = 1.625 in.
h = 4.25 in.
Calculating the surface area:
A = 2π(1.625)(1.625 + 4.25)
A ≈ 2π(1.625)(5.875)
A ≈ 60.97 square inches
Next, let's calculate the cost of the packaging for option A:
Cost_A = A * $0.00043
Cost_A ≈ 60.97 * $0.00043
Cost_A ≈ $0.0262621
Now, let's move on to option B, the rectangular prism container:
The formula to calculate the surface area of a rectangular prism is given by:
A = 2lw + 2lh + 2wh
where A is the surface area, l is the length, w is the width, and h is the height.
Using the given values:
l = 3.5 in.
w = 1.5 in.
h = 6.7 in.
Calculating the surface area:
A = 2(3.5)(1.5) + 2(3.5)(6.7) + 2(1.5)(6.7)
A = 25.9 square inches
Next, let's calculate the cost of the packaging for option B:
Cost_B = A * $0.00036
Cost_B = 25.9 * $0.00036
Cost_B ≈ $0.009324
Comparing the costs, we can see that option B is cheaper:
Cost_A ≈ $0.0262621
Cost_B ≈ $0.009324
Therefore, based on the cost of packaging, option B, the container shaped like a rectangular prism, is cheaper.