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Calculus

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A light is suspended at a height h above the floor. The illumination at the point P is inversely proportional to the square of the distance from the point P to the light and directly proportional to the cosine of the angle θ. How far from the floor should the light be to maximize the illumination at the point P? (Let d = 6 ft.)

  • Calculus -

    I guess d is from below the light to a point on the floor?
    If so:

    I = k cos T / (h^2+d^2)
    where cos T = h/(h^2+d^2)^.5
    so
    I = k h/(h^2+d^2)^1.5
    dI/dh = k [ (h^2+d^2)^1.5 -h(1.5)(h^2+d^2)^.5 (2h) ] / (h^2+d^2)^3
    = 0 for extreme
    (h^2+d^2)^1.5 = 3 h^2 (h^2+d^2)^.5
    (h^2+d^2) = 3 h^2
    d^2 = 2 h^2
    d = h sqrt 2
    h = 6/sqrt 2 = 4.24 ft

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