Q asked in Science & MathematicsMathematics · 4 months ago

Use polar coordinates to find the volume of the given solid. Above the cone z = sqrt x2 + y2 and below the sphere x2 + y2 + z2 = 49?

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  • Pope
    Lv 7
    4 months ago
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    You really, really need to work on your notation. This time I am going to help. I think these must be the two surfaces you intended:

    z = √(x² + y²)

    x² + y² + z² = 49

    Some indication of exponent is necessary for those 2s, and in order to describe a cone, that square root argument must include both terms. Use brackets.

    Both surfaces have symmetry on the z-axis. The radius of any level section of the conic surface is equal to z. The sphere has radius 7. The region you defined (sort of) is a conic sector.

    Find the height of the circle where cone meets sphere.

    z = r

    z = √(49 - z²)

    z = 7/√(2)

    Now I am looking at the spherical surface of the sector.

    Its height is 7 - 7/√(2) = [14 - 7√(2)]/2.

    Project that surface horizontally away from the z-axis, onto the vertical cylinder circumscribing the sphere. Its image is a right cylinder having these dimensions:

    radius = 7

    height = [14 - 7√(2)]/2

    area = 14π[14 - 7√(2)]/2 = 49[2 - √(2)]π

    That also is the area of the spherical surface of the sector. Volume of a sphere sector is proportional to the area of its spherical surface.

    (sector volume) : (sphere volume) = 49[2 - √(2)]π : (sphere surface area)

    (sector volume) : (4/3)(7³)π = 49[2 - √(2)]π : 4(7²)π

    sector volume = 343[2 - √(2)]π/3

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