Find the points on the surface y2 = 25 + xz that are closest to the origin.

Using lagrange multipliers, it is found that the points are: (0,-5,0) and (0,5,0).

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The distance from any point to the origin is:

Thus, we want to minimize:

The restriction is:

The gradients are:

The relations to build a system are:

The restriction is also considered.

From the second equation, , and then:

Which is only true for x = z = 0.

Then, in the restriction:

Thus, the points are: (0, -5, 0) and (0, 5, 0).

A similar problem is given at link

56.6

step-by-step explanation:

the ratio of the perimeters is equal to the scale.

the ratio of the areas is equal to the square of the scale.

34 / 20 = k

a / 19.6 = k²

substituting:

a / 19.6 = (34 / 20)²

a = 19.6 (34 / 20)²

a ≈ 56.6

its a trick question

step-by-step explanation:

dogs love water, just turn on the hose and it will run in circles chasing the water all day : )