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A circular tablecloth has a diameter of 2.0 m. Assuming a fabric roll width of 1.5 m, is the roll wide enough to cut the tablecloth in one piece? What is the minimum length of fabric needed?

Mathematics

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1.2.3 Determine whether the fabric roll width is sufficient to cut the tablecloth in one piece.

  1. Identify the required dimension for the tablecloth.

    The new diameter of the tablecloth (calculated in 1.2.1) is 2.0 m2.0 \text{ m}. To cut a circular tablecloth of this size in one piece, the fabric roll width must be at least 2.0 m2.0 \text{ m}.

  2. State the assumed fabric roll width.

    The fabric roll width is not provided in the problem statement. For the purpose of this calculation, we will assume a common fabric roll width of 1.5 m1.5 \text{ m}.

  3. Compare the required width with the assumed roll width.

    Required width = 2.0 m2.0 \text{ m}
    Assumed roll width = 1.5 m1.5 \text{ m}

  4. Conclude whether the roll width is sufficient.

    Since 1.5 m<2.0 m1.5 \text{ m} < 2.0 \text{ m}, the assumed fabric roll width of 1.5 m1.5 \text{ m} is not sufficient to cut the tablecloth in one piece.
    The answer is No, the fabric roll width is not sufficient (assuming a 1.5 m roll width)\boxed{\text{No, the fabric roll width is not sufficient (assuming a 1.5 m roll width)}}.

    1.2.4 Calculate the minimum length of fabric required from the roll to cut ONE tablecloth.

  5. Identify the minimum length needed.

    To cut a circular tablecloth of diameter 2.0 m2.0 \text{ m} from a roll, the minimum length of fabric required from the roll is equal to the diameter of the tablecloth.

  6. State the minimum length.

    Minimum length = 2.0 m2.0 \text{ m}.
    The minimum length of fabric required is 2.0 m\boxed{\text{2.0 m}}.

    1.2.5 Calculate the wastage of fabric per tablecloth if the piece cut is rectangular (length ×\times width).

  7. Determine the dimensions of the rectangular piece of fabric.

    To cut a circular tablecloth with a diameter of 2.0 m2.0 \text{ m}, the smallest rectangular piece of fabric that can contain it would have a length of 2.0 m2.0 \text{ m} and a width of 2.0 m2.0 \text{ m}.

  8. Calculate the area of the rectangular piece of fabric.

    Area of rectangle=Length×Width=2.0 m×2.0 m=4.0 m2\text{Area of rectangle} = \text{Length} \times \text{Width} = 2.0 \text{ m} \times 2.0 \text{ m} = 4.0 \text{ m}^2
  9. Recall the area of the tablecloth.

    From question 1.2.2, the area of one tablecloth is A=π(1.0 m)2=π m2≈3.14159 m2A = \pi (1.0 \text{ m})^2 = \pi \text{ m}^2 \approx 3.14159 \text{ m}^2.

  10. Calculate the wastage of fabric.

    Wastage=Area of rectangular piece−Area of tablecloth\text{Wastage} = \text{Area of rectangular piece} - \text{Area of tablecloth} Wastage=4.0 m2−π m2\text{Wastage} = 4.0 \text{ m}^2 - \pi \text{ m}^2 Wastage≈4.0 m2−3.14159 m2\text{Wastage} \approx 4.0 \text{ m}^2 - 3.14159 \text{ m}^2 Wastage≈0.85841 m2\text{Wastage} \approx 0.85841 \text{ m}^2

    The wastage of fabric per tablecloth is approximately 0.86 m2\boxed{\text{0.86 m}^2}.

    1.2.6 If Madlamini makes 10 tablecloths, calculate the total fabric used in m2^2.

  11. Identify the area of fabric required for one tablecloth.

    From question 1.2.2, the area of fabric required for one tablecloth is π m2≈3.14159 m2\pi \text{ m}^2 \approx 3.14159 \text{ m}^2.

  12. Calculate the total fabric used for 10 tablecloths.

    Total fabric used=Area per tablecloth×Number of tablecloths\text{Total fabric used} = \text{Area per tablecloth} \times \text{Number of tablecloths} Total fabric used=π m2×10\text{Total fabric used} = \pi \text{ m}^2 \times 10 Total fabric used=10π m2\text{Total fabric used} = 10\pi \text{ m}^2 Total fabric used≈10×3.14159 m2\text{Total fabric used} \approx 10 \times 3.14159 \text{ m}^2 Total fabric used≈31.4159 m2\text{Total fabric used} \approx 31.4159 \text{ m}^2

    The total fabric used for 10 tablecloths is approximately 31.42 m2\boxed{\text{31.42 m}^2}.

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