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Calculate which container has the largest volume. Give the volume to the nearest litre.

Mathematics

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3 parts
6.1

1571 l\text{1571 l}

6.2

7.68 m2\text{7.68 m}^2

7.85 m2\text{7.85 m}^2

6.3

3.84 m\text{3.84 m}

3.93 m\text{3.93 m}

Step-by-step solution

10 steps

Here are the calculations for the containers:

Part 6.1· 6 steps

Calculate which container has the largest volume. Give the volume to the nearest litre.

  1. Convert all dimensions to meters for consistency.

    For the rectangular container:
    Length (ll) = 80 cm=0.8 m80 \text{ cm} = 0.8 \text{ m}
    Width (ww) = 80 cm=0.8 m80 \text{ cm} = 0.8 \text{ m}
    Height (hh) = 2 m2 \text{ m}

    For the cylindrical container:
    Diameter (DD) = 1 m1 \text{ m}
    Radius (rr) = D2=1 m2=0.5 m\frac{D}{2} = \frac{1 \text{ m}}{2} = 0.5 \text{ m}
    Height (hh) = 2 m2 \text{ m}

  2. Calculate the volume of the rectangular container.

    Vrectangular=l×w×hV_{\text{rectangular}} = l \times w \times h Vrectangular=0.8 m×0.8 m×2 mV_{\text{rectangular}} = 0.8 \text{ m} \times 0.8 \text{ m} \times 2 \text{ m} Vrectangular=1.28 m3V_{\text{rectangular}} = 1.28 \text{ m}^3
  3. Convert the volume of the rectangular container to litres.

    Given 1000 cm3=1 l1000 \text{ cm}^3 = 1 \text{ l}, and 1 m3=(100 cm)3=1,000,000 cm31 \text{ m}^3 = (100 \text{ cm})^3 = 1,000,000 \text{ cm}^3.
    Therefore, 1 m3=1,000,0001000 l=1000 l1 \text{ m}^3 = \frac{1,000,000}{1000} \text{ l} = 1000 \text{ l}.

    Vrectangular=1.28 m3×1000lm3V_{\text{rectangular}} = 1.28 \text{ m}^3 \times 1000 \frac{\text{l}}{\text{m}^3} Vrectangular=1280 lV_{\text{rectangular}} = 1280 \text{ l}
  4. Calculate the volume of the cylindrical container.

    Vcylindrical=πr2hV_{\text{cylindrical}} = \pi r^2 h Vcylindrical=π×(0.5 m)2×2 mV_{\text{cylindrical}} = \pi \times (0.5 \text{ m})^2 \times 2 \text{ m} Vcylindrical=π×0.25 m2×2 mV_{\text{cylindrical}} = \pi \times 0.25 \text{ m}^2 \times 2 \text{ m} Vcylindrical=0.5π m3V_{\text{cylindrical}} = 0.5\pi \text{ m}^3

    Using π≈3.14159\pi \approx 3.14159:

    Vcylindrical≈0.5×3.14159 m3V_{\text{cylindrical}} \approx 0.5 \times 3.14159 \text{ m}^3 Vcylindrical≈1.570795 m3V_{\text{cylindrical}} \approx 1.570795 \text{ m}^3
  5. Convert the volume of the cylindrical container to litres and round to the nearest litre.

    Vcylindrical≈1.570795 m3×1000lm3V_{\text{cylindrical}} \approx 1.570795 \text{ m}^3 \times 1000 \frac{\text{l}}{\text{m}^3} Vcylindrical≈1570.795 lV_{\text{cylindrical}} \approx 1570.795 \text{ l} Vcylindrical≈1571 lV_{\text{cylindrical}} \approx 1571 \text{ l}
  6. Compare the volumes.

    1571 l>1280 l1571 \text{ l} > 1280 \text{ l}.
    The cylindrical container has the largest volume.

    The container with the largest volume is the cylindrical container, with a volume of 1571 l\boxed{\text{1571 l}}.

Part 6.2

7.68 m2\text{7.68 m}^2

Calculate the surface area of each container.

  1. Calculate the surface area of the rectangular container.

    SArectangular=2(lw+lh+wh)SA_{\text{rectangular}} = 2(lw + lh + wh) SArectangular=2((0.8 m×0.8 m)+(0.8 m×2 m)+(0.8 m×2 m))SA_{\text{rectangular}} = 2((0.8 \text{ m} \times 0.8 \text{ m}) + (0.8 \text{ m} \times 2 \text{ m}) + (0.8 \text{ m} \times 2 \text{ m})) SArectangular=2(0.64 m2+1.6 m2+1.6 m2)SA_{\text{rectangular}} = 2(0.64 \text{ m}^2 + 1.6 \text{ m}^2 + 1.6 \text{ m}^2) SArectangular=2(3.84 m2)SA_{\text{rectangular}} = 2(3.84 \text{ m}^2) SArectangular=7.68 m2SA_{\text{rectangular}} = 7.68 \text{ m}^2
  2. Calculate the surface area of the cylindrical container.

    SAcylindrical=2πr2+2πrhSA_{\text{cylindrical}} = 2\pi r^2 + 2\pi rh SAcylindrical=2π(0.5 m)2+2π(0.5 m)(2 m)SA_{\text{cylindrical}} = 2\pi (0.5 \text{ m})^2 + 2\pi (0.5 \text{ m})(2 \text{ m}) SAcylindrical=2π(0.25 m2)+2π(1 m2)SA_{\text{cylindrical}} = 2\pi (0.25 \text{ m}^2) + 2\pi (1 \text{ m}^2) SAcylindrical=0.5π m2+2π m2SA_{\text{cylindrical}} = 0.5\pi \text{ m}^2 + 2\pi \text{ m}^2 SAcylindrical=2.5π m2SA_{\text{cylindrical}} = 2.5\pi \text{ m}^2

    Using π≈3.14159\pi \approx 3.14159:

    SAcylindrical≈2.5×3.14159 m2SA_{\text{cylindrical}} \approx 2.5 \times 3.14159 \text{ m}^2 SAcylindrical≈7.853975 m2SA_{\text{cylindrical}} \approx 7.853975 \text{ m}^2

    The surface area of the rectangular container is 7.68 m2\boxed{\text{7.68 m}^2}.
    The surface area of the cylindrical container is 7.85 m2\boxed{\text{7.85 m}^2} (rounded to two decimal places).

Part 6.3

3.84 m\text{3.84 m}

Metal sheets, 2 m wide, are used to manufacture these containers. Calculate the length of sheeting required to make each container.

  1. Calculate the length of sheeting for the rectangular container.

    Length of sheeting = Total Surface Area / Width of sheet

    Lrectangular=SArectangularwidth of sheetL_{\text{rectangular}} = \frac{SA_{\text{rectangular}}}{\text{width of sheet}} Lrectangular=7.68 m22 mL_{\text{rectangular}} = \frac{7.68 \text{ m}^2}{2 \text{ m}} Lrectangular=3.84 mL_{\text{rectangular}} = 3.84 \text{ m}
  2. Calculate the length of sheeting for the cylindrical container.

    Lcylindrical=SAcylindricalwidth of sheetL_{\text{cylindrical}} = \frac{SA_{\text{cylindrical}}}{\text{width of sheet}} Lcylindrical=7.853975 m22 mL_{\text{cylindrical}} = \frac{7.853975 \text{ m}^2}{2 \text{ m}} Lcylindrical≈3.9269875 mL_{\text{cylindrical}} \approx 3.9269875 \text{ m} Lcylindrical≈3.93 mL_{\text{cylindrical}} \approx 3.93 \text{ m}

    The length of sheeting required for the rectangular container is 3.84 m\boxed{\text{3.84 m}}.
    The length of sheeting required for the cylindrical container is 3.93 m\boxed{\text{3.93 m}} (rounded to two decimal places).

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