The plane x + y + 2z = 12 intersects the paraboloid z = x2 +...

The plane

x + y + 2z = 12

intersects the paraboloid

z = x² + y²

in an ellipse. Find the points on the ellipse that are nearest to and farthest from the origin.

nearest point

(x, y, z)

=

farthest point

(x, y, z)

=

Expert Solution

Thus, the nearest point is (3/2,3/2,9/2) and the farthest point is (-2,-2,8).

milcah answered 2 years ago

The paraboloid z = 8 − x − x2 − 2y2 intersects the plane x =...

The paraboloid z = 8 − x − x2 − 2y2 intersects the plane x = 4 in a parabola. Find parametric equations in terms of t for the tangent line to this parabola at the point (4, 2, −20). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)

F(x,y,z) = x^2z^2i + y^2z^2j + xyz k S is the part of the paraboloid z...

F(x,y,z) = x^2z^2i + y^2z^2j + xyz k S is the part of the paraboloid z = x^2+y^2that lies inside the cylinder x^2+y^2 = 16, oriented upward.

The plane x+y+z= 24 intersects the cone x2+y2= z2 in an ellipse. The goal of this...

The plane x+y+z= 24 intersects the cone x2+y2= z2 in an ellipse. The goal of this exercise is to find the two points on this ellipse that are closest to and furthest away from the xy-plane. Thus, we want to optimize F(x,y,z)= z, subject to the two constraints G(x,y,z)= x+y+z= 24 and H(x,y,z)= x2+y2-z2= 0.

The plane y + z = 7 intersects the cylinder x2 + y2 = 41 in...

The plane y + z = 7 intersects the cylinder x2 + y2 = 41 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (4, 5, 2). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)

So we have a paraboloid x^2 + y^2 - 2 = z and the plane x...

So we have a paraboloid x^2 + y^2 - 2 = z and the plane x + y +z = 1 how do we find the center of mass? For some reason we have to assume the uniform density is 8? Seems complicated because I don't know where to start?

h Consider a solid T enclosed by the paraboloid z = x^2 +y^2 and the plane...

h Consider a solid T enclosed by the paraboloid z = x^2 +y^2 and the plane z = 4 (the solid above the paraboloid and below the plane). Let M the (closed) surface representing the boundary surface of T. The surface M consists of two surfaces: the paraboloid M1 and the lid M2. Orient M by an outward normal. Let F=(z,2y,-2) Compute the integral using the Divergence theorem. Carry out the computation of the triple integral using the spherical coordinates.

Find the surface area of the part of the paraboloid x = y^2 + z^2 that...

Find the surface area of the part of the paraboloid x = y^2 + z^2 that lies inside the cylinder y^2 + z^2 = 25

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e−x2...

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e−x2 − 3y2 − 7z2 where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P(2, −1, 2) in the direction towards the point (4, −4, 4). answer in °C/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

The temperature at a point (x, y, z) is given by T(x, y, z) = 300e−x2...

The temperature at a point (x, y, z) is given by T(x, y, z) = 300e−x2 − 3y2 − 9z2 where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P(4, −1, 3) in the direction towards the point (6, −2, 6) (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

Find the coordinates of the point (x, y, z) on the plane z = 4 x...

Find the coordinates of the point (x, y, z) on the plane z = 4 x + 1 y + 4 which is closest to the origin.

Question