Question 880354
The maximum value is {{{2*2*2*2*2=2^5=32}}} .
In two dimensions, for a fixed perimeter,
{{{length+width=perimeter/2}}} is fixed,
the largest rectangle we can make is a square,
which has the same measurement for both dimensions: length = width.
In three dimensions, if we have a maximum (or a fixed value) for {{{width+length+height}}} ,
and we are trying to make the cuboid box with the larges volume,
the best choice is a cube with {{{width=length=height}}} .
The same works for 5 positive numbers with a given sum:
the greatest product is obtained when all numbers are the equal.
So with {{{a=b=c=d=e}}} , {{{a+b+c+d+e=10}}} turns into
{{{a+a+a+a+a=10}}} ---> {{{5a=10}}} ---> {{{a=10/5}}} ---> {{{a=2}}} .
Then {{{a*b*c*d*e=2*2*2*2*2=2^5=32}}} .
The explanation your teacher wants depends on the level of your class.