Gfind the point on the plane x + 4y + 2z =14 closest to the point (1,1,1)

(4/3, 7/3, 5/3)

Step-by-step explanation:

The line perpendicular to the plane through the point will be ...

... (x, y, z) = (1, 1, 1) +t(1, 4, 2)

Using this expression for (x, y, z), we can find the point on the line where it intersects the plane. That point is the one of interest.

... (1 +t) +4(1 +4t) +2(1 +2t) = 14 . . . . . substituting for (x, y, z) in the equation of the plane

... 7 +21t = 14 . . . . . collect terms

... 21t = 7 . . . . . . . . subtract 7

... t = 7/21 = 1/3  . . . divide by 21

The point of interest is ...

... (x, y, z) = (1, 1, 1) + (1/3)(1, 4, 2) = (1 1/3, 2 1/3, 1 2/3)

d. it is a convex pentagon because it has five sides and none of the sides would extend into the inside of the polygon.

step-by-step explanation:

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: