Explanation:
Let's calculate the volume of space taken up by the roller towels, the total amount of soft paper used, and the amount of cardboard used to manufacture the cardboard cylinder step by step.
1. Volume of Space Taken up by the Roller Towels:
To find the volume of space taken up by the roller towels, we need to calculate the volume of the larger cylinder (formed by the roller towels and cardboard) and then subtract the volume of the hollow cardboard cylinder.
Given:
Diameter of the larger cylinder = 10 cm (radius = 5 cm)
Height of the larger cylinder = 28 cm
Volume of larger cylinder = π * r² * h
Volume of larger cylinder = π * (5 cm)² * 28 cm
Volume of larger cylinder ≈ 2200 cm³
Now, let's calculate the volume of the hollow cardboard cylinder:
Given:
The distance between the outer and inner circumferences (thickness of the cardboard) = 3 cm
Volume of hollow cardboard cylinder = π * (radius of larger cylinder - thickness of cardboard)² * height of the larger cylinder
Volume of hollow cardboard cylinder = π * (5 cm - 3 cm)² * 28 cm
Volume of hollow cardboard cylinder ≈ 502 cm³
Now, we can find the volume of space taken up by the roller towels:
Volume of space taken up by the roller towels = Volume of larger cylinder - Volume of hollow cardboard cylinder
Volume of space taken up by the roller towels = 2200 cm³ - 502 cm³
Volume of space taken up by the roller towels ≈ 1698 cm³
2. Total Amount of Soft Paper Used:
Each roller towel has dimensions of 275 mm x 220 mm. To calculate the total amount of soft paper used, we need to convert these dimensions to the same units as the volume.
Given:
Width of each roller towel = 275 mm (convert to cm by dividing by 10)
Length of each roller towel = 220 mm (convert to cm by dividing by 10)
Area of one roller towel = Width * Length
Area of one roller towel = (275 cm / 10) * (220 cm / 10) = 27.5 cm * 22 cm = 605 cm²
Total area of 52 roller towels = 52 * 605 cm² ≈ 31460 cm²
3. Amount of Cardboard Used for the Cardboard Cylinder:
To calculate the amount of cardboard used for the cardboard cylinder, we need to find the surface area of the cylinder.
Given:
Radius of larger cylinder (formed by the roller towels and cardboard) = 5 cm
Height of the larger cylinder = 28 cm
Surface area of the cylinder = 2πrh + 2πr²
Surface area of the cylinder = 2π * 5 cm * 28 cm + 2π * (5 cm)²
Surface area of the cylinder ≈ 880 cm²
Therefore, approximately 880 cm² of cardboard is used to manufacture the cardboard cylinder.
In conclusion:
- The volume of space taken up by the roller towels is approximately 1698 cm³.
- The total amount of soft paper used is approximately 31460 cm².
- The amount of cardboard used to manufacture the cardboard cylinder is approximately 880 cm².