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52 rectangular sheets of soft paper called roller towels are joined together and wrapped around a hollow cardboard cylinder. The dimensions of each roller towel are 275 mmx220 mm. The roller towels and cardboard cylinder form a larger cylinder with a diameter of 10 cm and height 28 cm. The distance between the outer and inner circumferences is 3 cm.

Calculate the volume of space taken up by the roller towels. Calculate the total amount of soft paper used. Calculate the amount of cardboard used to manufacture the cardboard cylinder. ​

User Thedan
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1 Answer

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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.

User Carlos Quintanilla
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