# Thread: [SOLVED] Planes &amp; Archery

1. ## [SOLVED] Planes &amp; Archery

In my calculus and vectors course, I have been assigned the following problem:

In an archery challenge an arrow is fired from the point (3,4,-1) and is aimed at the plane x+y+z-30=0 such that x,y,z are greater than or equal to 0. Each shot is given a score by taking the point where the arrow hit the plane and multiplying the 3 coordinates together.
a) If a player shoots the arrow along a direction of [1,1,1] determine the score for their shot (I have solved this one, the answer being (11)(12)(7)=924)
b) Determine the direction that the player should aim in order to maximize their score.

I need help with (b). any help would be greatly appreciated!

2. Hello, Jenna!

In an archery challenge an arrow is fired from the point (3,4,-1)
and is aimed at the plane $\displaystyle x+y+z-30\:=\:0$ such that $\displaystyle x,y,z \geq 0$

Each shot is given a score by taking the point where the arrow hit the plane
and multiplying the three coordinates together.

a) If a player shoots the arrow along a direction of [1,1,1]
determine the score for their shot.
(I have solved this one, the answer being (11)(12)(7)=924) . I agree!

b) Determine the direction that the player should aim in order to maximize their score.
We want three nonnegative numbers, $\displaystyle x,y,z$, whose sum is 30
. . and whose product is a maximum.

We can use Lagrange multipliers to maximize: .$\displaystyle P \:=\:xyz$
. . subject to the constraint: .$\displaystyle x + y + z \:=\:30$

Or we can recall that, if a set of numbers has a fixed sum,
. . their product is a maximum when they are equal.

Either way, the archer should aim at $\displaystyle (10,10,10).$

3. We want three nonnegative numbers, , whose sum is 30
. . and whose product is a maximum.

We can use Lagrange multipliers to maximize: .
. . subject to the constraint: .

Or we can recall that, if a set of numbers has a fixed sum,
. . their product is a maximum when they are equal.

Either way, the archer should aim at

ohhh that makes plenty of sense! thank you very much! just one thing I'm a bit confuesed about.. that (10, 10, 10), would that be my actual direction vector? or is it simply the point that the archerer is aiming at? Or are those actually the same thing?