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math (solid mensuration)

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A light is placed 5 ft. from the center of a globe 3 ft. in diameter. Find the area of the illuminated portion.

thank you!

  • math (solid mensuration) -

    make a side view of the event
    draw a line from the light to the centre of the sphere,
    draw a line to the sphere representing your line of sight, this would be a tangent to the sphere
    It is easy to see, that this tangent would be 4 ft
    (the 3-4-5 right-angled triangle)
    the angle Ø at the centre of that triangle is sin^-1 (4/5) = appr 53.13°

    so we need the surface area of the "cap".
    let x be the distance from centre to base of the cap
    cos 53.13 = x/3
    .6 = x/3
    x = 1.8
    so the height , h, of the cap is 3-1.8 = 1.2

    The curved surface area of the spherical cap is 2πrh
    where r is the radius of the sphere and h is the height of the cap

    SA = 2π(3)(1.2) = 7.2π square feet

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