49.) Find the minimum distance from the point (2, 1, 4) to x + y + z = 4 by using the LaGrange method to minimize the (distance)².

Q: 4. Find the equations of the line tangent to the intersection of the surfaces z = x´ + 2y and

A:

Q: Find the point on the planez = x + y + 1closest to the point P = (1, 0, 0). Hint: Minimize the…

A: Let Q(x,y,z) be the point closest to the point P. Let D be the distance between these two points.…

Q: Find the minimum distance from the point to the plane x-y+z=3 at (1,-3,2)

A: We will find the minimum distance from the point to the plane x-y+z=3 at (1,-3,2) in the following…

Q: Construct the function d(x, y) giving the distance from a point (x, y, z) on the paraboloid z = 64 –…

A:

Q: Use the method of Lagrange multipliers to find the minimum distance from the origin (0, 0) to the…

A: We need to use the method of Lagrange's multiplier to determine the minimum distance from the point:…

Q: Minimize ?=?+3?+10 subject to:  2?+4?≥8  4?+3?≥12  ?,?≥0 Corner points at…

A: The objective function is given as, Zx,y=x+3y+10 Subjected to the following constraints,…

Q: Find the minimum distance from the point to the plane x - y + z = 3.

A: Now, formula of shortest distance of the plane ax+by+cz=d passing through the point (l, m, n) is,

Q: 5. What is the maximum vertical distance between the line y = x + 2 and the parabola y = x² for – 1…

A: The given equations are, y = x + 2       ....(1)  y = x2            ....(2) To find the maximum…

Q: Find the average squared distance between the origin and the points on the paraboloid z = 4 – x² –…

A:

Q: 1. Find where f(x, y) = 4x² + 8y? – 4x – 8y +5 has absolute extreme values over the square with…

A: To calculate the extreme values of f(x,y)bounded by the square, the following calculations need to…

Q: Find the minimum distance from the point (4, 0, 6) to the plane x − y + z = 3.

A: Find the minimum distance from the point (4, 0, 6) to the plane x − y + z = 3.

Q: 2. Use the method of Lagrange Multipliers to determine which point on the surface !++ = 1, æ, y, z >…

A: According to the Legrange multipliers method to find closest point on surface to origin, solve for…

Q: 5. Find the area and centroidal distances of the curve that lies between he equations y%3D4-x^2 and…

A: The region bounded by curves y=4-x2andy=x+2 To find the area and centroidal distances.

Q: (6) Find the orthogonal trajectories of the family of curves r 4C cos 0, where C being %3D…

A: We have to solve given problem:

Q: 3. Use the method of Lagrange multipliers to find the minimum distance between the paraboloid z = x²…

A: (3)Let (x,y,z) lies on parabolaThe distance d=x-32+y2+z2such that z=x2+4y2Lagrabge multiplier…

Q: 3. Use the method of Lagrange multipliers to find the minimum distance between the paraboloid z = x?…

A: Given,                               z=x2+4y2and the point (3, 0, 0).

Q: find the point on the line y=2x+10 that is closest to the orgin, (0,0) What is the minimum distance?

A:

Q: (6) Find the orthogonal trajectories of the family of curves r= 2C cos 0, where C being parameter.

A: Consider the given curve. r=2Ccosθ          (1) Find the derivative of the given equation.…

Q: SET UP the necessary equations to find the point on the intersection of the surfaces x2 + 4y2 + 4z2…

A: The solution are next step

Q: 13. Show that the length of the portion of any tangent line to the ball x2/3 + y2/3 cut off by the…

A: Given: The equation of the ball is: x23+y23=a23...........(1) We have to show that the length of the…

Q: 2. Use the method of Lagrange multipliers to show that for x, y ≥ 0, we have z"+"> (²+)" 2

A: Introduction: For minimizing a multi-variable function under some certain constraint or constraints,…

Q: Consider a circle of radius 2 centered at the origin. Use the Lagrange multiplier method to find the…

A: It is given that circle is of radius 2 and centered at origin. The function isfx,y=3x-y+5.

Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 <=t<= 2pai closest to the point (3/4,…

A:

Q: · What is the shortest distance from the surface xy + 6x + z = 41 to the origin? distance =

A: The surface is given as , xy+6x+z2=41 find the shortest distance from the surface to the origin.

Q: Consider the inmplicitly defined curve xy +3x2 4. %3D O Verify that the point (1, 1) is on the…

A: Consider, the implicitely defined curve                          xy2+3x2=4 We have to verify the…

Q: Let L be the line with equation y = x – 1. Find the minimum distance between L and the point (4, 5)…

A: In order to use method of Lagrange multiplier, we first need to find the function to be minimized.…

Q: What is the shortest distance from the surface xy +12x +z^2 =144 to the origin?

A: To find the shortest distance from surface xy +12x +z2 =144  to origin.

Q: 1. Find the smallest distance from the point (0, c, 0) to the surface y = x² + z².

A:

Q: Q1: Use ewton- Raphson method to estimate the Yoot of fcx) = x -x -1 , emp loying an initial %3D…

A:

Q: 2. Find the absolute maximum and minimum values of f(x, y) = 4x+6y-(r²+y?) on the rectangular region…

A:

Q: Find the maximum and minimum values attained by f(x, y, z) = 2xyz on the unit ball x2 + y2 + z2 < 1.…

A: fx,y,z=2xyz on the unit ball x2+y2+z2≤1 gx,y,z=x2+y2+z2-1 ∇f=∂f∂x,∂f∂y,∂f∂z ∇g=2yz,2xz,2yx…

Q: Solve the given optimization problem by using substitution. Minimize S- xy + xz + yz subject to xyz…

A: S=xy+xz+yz  =xyzz+xyzy+xyzx

Q: 3 Find the Minimum distance from the origin to the plane X+2y-z=5

A:

Q: . Use the method of Lagrange multipliers to show that for x, y > 0, we have a" +y" > (팔1)" 2

A: We will first try to minimise (xn + yn)/2 using Lagrange multipliers then show that its minimum…

Q: Use the method of Lagrange multipliers to minimize f (xy) =x² +y² on the hyperbola x-y=1.

A: Given: fx,y=x2+y2 and xy=1

Q: By minimizing the function ƒ(x, y, u, y) = (x - u)2 + (y - y)2 subject to the constraints y = x + 1…

A:

Q: What function should be used to minimize the distance between y = x + 5 and the point (3, 1)? O D(æ)…

A: Explanation of the solution is given below...

Q: 5 Find the valume of the Sotids abtaind by-revelving the Yegion in the firstquadrand boundby the…

A: Sketch the graph and find the region. Using shell method, volume is evaluated.

Q: 6. Find the maxium andd minimum values of the following functions by using Lagrange Multiplier. a)…

A:

Q: 4. (a) Use the trapezoidal rule to numerically integrate from a = 0 to b= 2 /(z)-0.2+25 +3r

A:

Q: Find the minimum distance from the point (0,0,0) to the plane 9x+4y+z=5.

A:

Q: Use the Improved Euler Method to estimate the values of the function y, solution of the imutial…

A:

Q: Find the minimum distance from the cone z = √(x2 + y2) to the point (-6, 4, 0).

A: To find the minimum distance

Q: 2 : Use Lagrange multipliers method to show that the maximum value of f(x, y, z) = x²y²z² on the…

A: In lagrange method, the Critical points are the points that cancel the first order partial…

Q: 3. Use the method of Lagrange multipliers to find the minimum distance between the paraboloid 2= z +…

A:

Q: 4 Determine the surface area of a funnel that is generated by revolving the graph of 1 y = f(x) =…

A: Given  y=fx=x3+112xand interval 1,2

Q: Find two numbers x and y with x-y = 20 for which the term xy is minimized.

A: In this problem, we have to find the two numbers "x and y"

Q: Find the minimum distance from the cone z= to the point (6, 6, 0).

A:

Q: What is the largest vertical distance between the graphs of the functions y = Vx and y = x² in the…

A:

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.