Find the amount of water displaced by a solid spherical ball of diameter

General
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find the amount of water displaced by a solid spherical ball of diameter

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Find the amount of water displaced by a solid spherical ball of diameter

Answer: To find the amount of water displaced by a solid spherical ball, you need to determine the volume of the sphere. The volume of the sphere will be equal to the volume of water displaced when the sphere is fully submerged.

Formula for the Volume of a Sphere:

The volume ( V ) of a sphere can be calculated using the formula:

V = \frac{4}{3} \pi r^3

where:

  • ( r ) is the radius of the sphere.
  • ( \pi ) is a constant approximately equal to 3.14159.

Steps to Calculate the Volume:

  1. Determine the Radius:

    • The diameter ( d ) of the sphere is given.
    • The radius ( r ) is half of the diameter.
    r = \frac{d}{2}

  2. Calculate the Volume:

    • Substitute the radius ( r ) into the volume formula.
    V = \frac{4}{3} \pi \left( \frac{d}{2} \right)^3
    • Simplify the expression:
    V = \frac{4}{3} \pi \left( \frac{d^3}{8} \right)
    V = \frac{\pi d^3}{6}

Example Calculation:

Let’s assume the diameter ( d ) of the spherical ball is 10 cm.

  1. Determine the Radius:

    r = \frac{10 \, \text{cm}}{2} = 5 \, \text{cm}

  2. Calculate the Volume:

    V = \frac{4}{3} \pi (5 \, \text{cm})^3
    V = \frac{4}{3} \pi (125 \, \text{cm}^3)
    V = \frac{500}{3} \pi \, \text{cm}^3
    V \approx 523.6 \, \text{cm}^3 \, (\text{using} \, \pi \approx 3.14159)

Therefore, a solid spherical ball with a diameter of 10 cm will displace approximately 523.6 cm³ of water when fully submerged.

Conclusion:

The amount of water displaced by a solid spherical ball is equal to the volume of the sphere, which can be calculated using the formula ( V = \frac{\pi d^3}{6} ). By substituting the given diameter into this formula, you can determine the volume of water displaced.