Let x, y, and z be real numbers such that x^2 + y^2 + z^2 = 1. Find the maximum value of 9x+12y+8z.?
1 Answer
May 21, 2017
Explanation:
The equation:
x^2+y^2+z^2=1
describes the unit sphere in
The equation:
9x+12y+8z = k
describes a plane with normal vector
The unit vector in the same direction is given by dividing by:
||<9, 12, 8>|| = sqrt(9^2+12^2+8^2)
color(white)(||<9, 12, 8>||) = sqrt(81+144+64)
color(white)(||<9, 12, 8>||) = sqrt(289)
color(white)(||<9, 12, 8>||) = 17
that is:
< 9/17, 12/17, 8/17 >
This normal vector will intersect the unit sphere at the point:
(9/17, 12/17, 8/17)
This point will be the intersection of the plane and the unit sphere if the plane just touches the unit sphere - that is when
Then we find:
9x+12y+8z = 9(9/17)+12(12/17)+8(8/17)
color(white)(9x+12y+8z) = (9^2+12^2+8^2)/17 = 289/17 = 17