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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. ?

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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)

• 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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