11. [-/5 Points] DETAILS Find the point on the sphere x^2 +...

Transcript

00:01 In this problem, we are asked to find the point on the sphere x square plus y, square plus z square is equal to 16, that is farthest from the point 111.

00:15 Now, let x, y, z be any point on this sphere.

00:19 Then by distance formula, we have the distance d from this point to the point 1111 is square root of x minus 1 the whole square.

00:31 Plus y minus 1 the whole square plus z minus 1 the whole square now taking square on both sides we get d square is equal to x minus 1 the whole square plus y minus 1 the whole square plus z minus 1 the whole square now if we take f of x y of x y set to be the function d square that is x minus 1 the whole square plus y minus 1 the whole square plus z minus 1 the whole square to find the father's point we need to maximize this function f of x y said subject to the constraint x square plus y square plus is that square is equal to 16 now this can be done by the method of legrangea multipliers so if lambda is a legrangea multiplier then we must have del f is equal to lambda del g we can solve this in order to find the maximum value of this function and thereby find the point where the maximum mark up so we have del f is the gradient of the function f and that is the vector formed by the components do f by do x do y do x by do i said and that must be equal to del lambda times del g where g of x y said is the function x square plus y square plus is set square now del g is do g by do x x by do y so that we get do i.

02:27 X is the partial derivative of f with respect to x and that is two times x minus one similarly do i.

02:33 By do y is two times y minus one and da f by do i so is it is two times z minus one and this must be equal to lambda times del g is 2x 2y to z now from here by equating the corresponding components we get two times x minus 1 is equal to two lambda x 2 times y minus 1 is equal to 2 lambda y and 2 times z minus 1 is equal to 2 lambda set so from here we get x minus lambda x is equal to 1 that is is we get the value of x to be 1 divided by 1 minus lambda similarly from the second equation we get y is equal to 1 divided by 1 minus lambda and from the last equation we get z is equal to 1 divided by 1 minus lambda now substituting in the equation x square plus y square plus z square is equal to 16, we get 1 by 1 minus lambda the whole square plus 1 by 1 minus lambda the whole square plus 1 by 1 minus lambda the whole square is equal to 16...

Question

11. [-/5 Points] DETAILS Find the point on the sphere $x^2 + y^2 + z^2 = 400$ that is farthest from the point $(-20, 13, 6)$. (\qquad , \qquad , \qquad )

Question

11. [-/5 Points] DETAILS Find the point on the sphere $x^2 + y^2 + z^2 = 400$ that is farthest from the point $(-20, 13, 6)$. (\qquad , \qquad , \qquad )

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