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The radius of a sphere is 10 cm. Find its surface area and volume. 2. The diameter of an iron ball is 6 cm. Find its surface area and volume (use pi = 3.14).

Mathematics

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1. The radius of a sphere is 10 cm. Find its surface area and volume.

  1. Identify the given radius and the formulas for surface area and volume of a sphere.

    Given: r=10 cmr = 10 \text{ cm}.
    Surface Area (SASA) = 4πr24\pi r^2
    Volume (VV) = 43πr3\frac{4}{3}\pi r^3

  2. Calculate the surface area.

    SA=4π(10 cm)2SA = 4\pi (10 \text{ cm})^2 SA=4π(100 cm2)SA = 4\pi (100 \text{ cm}^2) SA=400π cm2SA = 400\pi \text{ cm}^2

    Using π≈3.14159\pi \approx 3.14159:

    SA≈400×3.14159 cm2SA \approx 400 \times 3.14159 \text{ cm}^2 SA≈1256.64 cm2SA \approx 1256.64 \text{ cm}^2
  3. Calculate the volume.

    V=43π(10 cm)3V = \frac{4}{3}\pi (10 \text{ cm})^3 V=43π(1000 cm3)V = \frac{4}{3}\pi (1000 \text{ cm}^3) V=40003π cm3V = \frac{4000}{3}\pi \text{ cm}^3

    Using π≈3.14159\pi \approx 3.14159:

    V≈40003×3.14159 cm3V \approx \frac{4000}{3} \times 3.14159 \text{ cm}^3 V≈1333.33×3.14159 cm3V \approx 1333.33 \times 3.14159 \text{ cm}^3 V≈4188.79 cm3V \approx 4188.79 \text{ cm}^3

    The surface area is 400\picm2 or approximately 1256.64 cm2\boxed{\text{400\pi cm}^2 \text{ or approximately 1256.64 cm}^2} and the volume is 40003π cm3 or approximately 4188.79 cm3\boxed{\text{}\frac{4000}{3}\pi \text{ cm}^3 \text{ or approximately 4188.79 cm}^3}.

    2. The diameter of an iron ball is 6 cm. Find its surface area and volume (use π=3.14\pi = 3.14).

  4. Identify the given diameter and calculate the radius.

    Given: d=6 cmd = 6 \text{ cm}.
    Radius r=d2=6 cm2=3 cmr = \frac{d}{2} = \frac{6 \text{ cm}}{2} = 3 \text{ cm}.
    Use π=3.14\pi = 3.14.

  5. Calculate the surface area.

    SA=4πr2SA = 4\pi r^2 SA=4×3.14×(3 cm)2SA = 4 \times 3.14 \times (3 \text{ cm})^2 SA=4×3.14×9 cm2SA = 4 \times 3.14 \times 9 \text{ cm}^2 SA=12.56×9 cm2SA = 12.56 \times 9 \text{ cm}^2 SA=113.04 cm2SA = 113.04 \text{ cm}^2
  6. Calculate the volume.

    V=43πr3V = \frac{4}{3}\pi r^3 V=43×3.14×(3 cm)3V = \frac{4}{3} \times 3.14 \times (3 \text{ cm})^3 V=43×3.14×27 cm3V = \frac{4}{3} \times 3.14 \times 27 \text{ cm}^3 V=4×3.14×9 cm3V = 4 \times 3.14 \times 9 \text{ cm}^3 V=113.04 cm3V = 113.04 \text{ cm}^3

    The surface area is 113.04 cm2\boxed{\text{113.04 cm}^2} and the volume is 113.04 cm3\boxed{\text{113.04 cm}^3}.

    3. Find the formula for the surface area and volume of a sphere in terms of its diameter dd.

  7. Relate the radius (rr) to the diameter (dd).

    r=d2r = \frac{d}{2}
  8. Substitute r=d2r = \frac{d}{2} into the surface area formula (SA=4πr2SA = 4\pi r^2).

    SA=4π(d2)2SA = 4\pi \left(\frac{d}{2}\right)^2 SA=4π(d24)SA = 4\pi \left(\frac{d^2}{4}\right) SA=πd2SA = \pi d^2

    The formula for surface area in terms of diameter is \boxed{\text{SA = \pi d^2}}.

  9. Substitute r=d2r = \frac{d}{2} into the volume formula (V=43πr3V = \frac{4}{3}\pi r^3).

    V=43π(d2)3V = \frac{4}{3}\pi \left(\frac{d}{2}\right)^3 V=43π(d38)V = \frac{4}{3}\pi \left(\frac{d^3}{8}\right) V=4πd324V = \frac{4\pi d^3}{24} V=16πd3V = \frac{1}{6}\pi d^3

    The formula for volume in terms of diameter is \boxed{\text{V = \frac{1}{6}\pi d^3}}.

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