The volume of a cone is 253 π cm3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?

1. 252 π cm
2. 256 π cm
3. 25 π cm3
4. 503 π cm3

1 answer

To solve the problem, we need to find the radius of the cone using its volume and then use that radius to find the volume of the sphere.

  1. Volume of the cone: The formula for the volume of a cone is given by:

    \[ V = \frac{1}{3} \pi r^2 h \]

    where \( r \) is the radius and \( h \) is the height of the cone.

    We know from the problem that the volume of the cone is \( 253 \pi \) cm³, so we set up the equation:

    \[ \frac{1}{3} \pi r^2 h = 253 \pi \]

    By dividing both sides by \( \pi \), we get:

    \[ \frac{1}{3} r^2 h = 253 \]

    Reorganizing gives us:

    \[ r^2 h = 759 \quad (1) \]

  2. Relationship between height and radius: According to the problem, the height of the cone is equal to the diameter of the sphere. Since the diameter of the sphere is \( 2r \), we can state:

    \[ h = 2r \quad (2) \]

  3. Substituting (2) into (1): We can now substitute \( h = 2r \) into our expression from step (1):

    \[ r^2 (2r) = 759 \]

    This simplifies to:

    \[ 2r^3 = 759 \]

    Dividing both sides by 2, we find:

    \[ r^3 = 379.5 \]

    Now we will calculate \( r \):

    \[ r = (379.5)^{1/3} \approx 7.25 \text{ cm} \]

  4. Finding the volume of the sphere: The volume of a sphere is given by the formula:

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

    Using \( r^3 = 379.5 \):

    \[ V = \frac{4}{3} \pi (379.5) \]

    Calculating this, we get:

    \[ V = \frac{4 \times 379.5}{3} \pi = \frac{1518}{3} \pi = 506 \pi , \text{cm}^3 \]

  5. Conclusion: The volume of the sphere is \( 506 \pi , \text{cm}^3 \). The closest option listed is:

None of the options in the provided list match. Please check the answer options again or confirm the conditions of the problem.

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