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The dimensions of the cylinder are:

Mathematics

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Here are the calculations for the volume and surface area of the cylinder.

The dimensions of the cylinder are:

  • Height (hh) = 10 dm10 \text{ dm}
  • Radius (rr) = 4 cm4 \text{ cm}
  1. Convert all dimensions to a consistent unit.

    We will convert decimeters to centimeters, as the radius is given in centimeters.

    1 dm=10 cm1 \text{ dm} = 10 \text{ cm} h=10 dm×10 cm1 dm=100 cmh = 10 \text{ dm} \times \frac{10 \text{ cm}}{1 \text{ dm}} = 100 \text{ cm}

    So, the height is 100 cm100 \text{ cm} and the radius is 4 cm4 \text{ cm}.

  2. Calculate the volume of the cylinder.

    The formula for the volume of a cylinder is V=πr2hV = \pi r^2 h.

    V=π(4 cm)2(100 cm)V = \pi (4 \text{ cm})^2 (100 \text{ cm}) V=π(16 cm2)(100 cm)V = \pi (16 \text{ cm}^2) (100 \text{ cm}) V=1600π cm3V = 1600 \pi \text{ cm}^3

    Using π≈3.14159\pi \approx 3.14159:

    V≈1600×3.14159 cm3V \approx 1600 \times 3.14159 \text{ cm}^3 V≈5026.55 cm3V \approx 5026.55 \text{ cm}^3

    The volume of the cylinder is 5026.55 cm3\boxed{\text{5026.55 cm}^3}.

  3. Calculate the surface area of the cylinder.

    The formula for the total surface area of a cylinder is SA=2πr2+2πrhSA = 2 \pi r^2 + 2 \pi r h.

    SA=2π(4 cm)2+2π(4 cm)(100 cm)SA = 2 \pi (4 \text{ cm})^2 + 2 \pi (4 \text{ cm}) (100 \text{ cm}) SA=2π(16 cm2)+2π(400 cm2)SA = 2 \pi (16 \text{ cm}^2) + 2 \pi (400 \text{ cm}^2) SA=32π cm2+800π cm2SA = 32 \pi \text{ cm}^2 + 800 \pi \text{ cm}^2 SA=832π cm2SA = 832 \pi \text{ cm}^2

    Using π≈3.14159\pi \approx 3.14159:

    SA≈832×3.14159 cm2SA \approx 832 \times 3.14159 \text{ cm}^2 SA≈2613.81 cm2SA \approx 2613.81 \text{ cm}^2

    The surface area of the cylinder is 2613.81 cm2\boxed{\text{2613.81 cm}^2}.

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