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Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

Of all the points on the plane 3x + 4y + z = 52, find the one that is closest to the origin. ?

2 Answers

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  • 2 months ago
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    Starting with the equation of the distance between two points in 3D-space:

    d = √[(x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)²]

    Using the points:

    (x, y, z) and (0, 0, 0), we get:

    d = √[(x - 0)² + (y - 0)² + (z - 0)²]

    d = √(x² + y² + z²)

    We are given this plane:

    3x + 4y + z = 52

    If we solve this for z in terms of x and y:

    z = 52 - 3x - 4y

    Then we can substitute this into the distance equation:

    d = √(x² + y² + z²)

    d = √[x² + y² + (52 - 3x - 4y)²]

    expand and simplify:

    d = √(x² + y² + 2704 - 156x - 208y - 156x + 9x² + 12xy - 208y + 12xy + 16y²)

    d = √(10x² + 17y² + 24xy - 312x - 416y + 2704)

    Now we have distance in terms of two unknowns.  If we get the partial derivatives for x and y, set both equal to zero, we'll have a system of two equations and two unknowns that you can solve for to find your x and y.  Then you can use those to work back and solve for z:

    We will need the chain rule in both.  first doing the partial with x, then the partial with y:

    d = √u and u = 10x² + 17y² + 24xy - 312x - 416y + 2704

    dd/du = 1/(2√u) and du/dx = 20x + 24y - 312 and du/dy = 34y + 24x - 416

    dd/dx = dd/du * du/dx and dd/dy = dd/du * du/dy

    dd/dx = 1/(2√u) * (20x + 24y - 312) and dd/dy = 1/(2√u) * (34y + 24x - 416)

    dd/dx = (20x + 24y - 312) / (2√u) and dd/dy = (34y + 24x - 416) / (2√u)

    dd/dx = (10x + 12y - 156) / √u and dd/dy = (17y + 12x - 208) / √u

    dd/dx = (10x + 12y - 156) / √(10x² + 17y² + 24xy - 312x - 416y + 2704) and

    dd/dy = (17y + 12x - 208) / √(10x² + 17y² + 24xy - 312x - 416y + 2704)

    Now set them both equal to zero:

    0 = (10x + 12y - 156) / √(10x² + 17y² + 24xy - 312x - 416y + 2704) and

    0 = (17y + 12x - 208) / √(10x² + 17y² + 24xy - 312x - 416y + 2704)

    Multiply both sides by the denominator:

    0 = 10x + 12y - 156 and 0 = 17y + 12x - 208

    There is the system of two equations and two unknowns.  Solving:

    -10x = 12y - 156

    x = -1.2y + 15.6

    0 = 17y + 12x - 208

    0 = 17y + 12(-1.2y + 15.6) - 208

    0 = 17y - 14.4y + 187.2 - 208

    0 = 2.6y - 20.8

    -2.6y = -20.8

    y = 8

    Now we can solve for x, then z:

    x = -1.2y + 15.6

    x = -1.2(8) + 15.6

    x = -9.6 + 15.6

    x = 6

    z = 52 - 3x - 4y

    z = 52 - 3(6) - 4(8)

    z = 52 - 18 - 32

    z = 2

    The closest point to the origin is: (6, 8, 2)

  • 2 months ago

    We are looking for min r were r = sqrt((x - 0)^2 + (y - 0)^2 + (x - 0)^2); so we take the partial differentials dr/dx, dr/dy, and dr/dz (sorry can't do the Greek letters) and set them = 0.  That gives us three equations and three unknowns that we can then solve for x, y, z which is the closet point p(x,y,z).

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