Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
The minimum distance between the origin and the plane 3x + y + 2z = 28, using the Lagrange Multipliers is 8.73 units.
In the question, we are asked to use Lagrange multipliers to find the minimum distance between the origin and the plans 3x + y + 2z = 28.
Lagrange multipliers state that if we want to minimize a function, f(x, y, z), subject to a constraint g(x, y, z) = constant, then ∇f = λ∇g.
We want to minimize the distance, and the distance formula says that the distance from the origin to a point (x, y, z), is given as:
D = √(x² + y² + z²), which can also be shown as:
D² = x² + y² + z².
The gradient of this function is: (2x + 2y + 2z), which needs to be equal to the gradient of the constraint equation multiplied by the constant, λ, that is, λ(3, 1, 2), which gives:
2x = 3λ, or, x = (3/2)λ,
2y = λ, or, y = λ/2, and,
2z = 2λ, or, z = λ.
Substituting these in the equation of the plane, 3x + y + 2z = 28, we get:
3(3/2)λ + λ/2 + λ = 28,
or, 6λ = 28,
or, λ = 28/6 = 14/3.
Thus, we get:
x = (3/2)λ = 7,
y = λ/2 = 7/3, and,
z = λ = 14/3.
Substituting these in the distance formula, we get:
D = √(7² + (7/3)² + (14/3)²) = √(49 + 49/9 + 196/9) = √(686/9) = 8.73.
Thus, the minimum distance between the origin and the plane 3x + y + 2z = 28, using the Lagrange Multipliers is 8.73 units.
Learn more about the Lagrange Multipliers at
https://brainly.com/question/15019779
#SPJ1
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.