Answer:
Length = Width = Height =36 inches
Volume =46,656 cubic inches
Explanation:
Let
x ----> the length of the box-shaped in inches
y ----> the width of the box-shaped in inches
z ---> the height of the box shaped in inches
we know that
----> equation A
Remember that
we have a square based box
so
----> equation B
substitute equation B in equation A

----> equation C
The volume of the box is equal to
----> equation D
substitute equation B and equation C in equation D

solve for x

Since we're looking for a maximum, that will happen when the slope of the above equation is 0. And the first derivative will give us that slope.
so
calculate the first derivative

equate to zero

solve for x
Factor -6x

The solutions are
x=0, x=36 in
Find the value of y

so

Find the value of z


therefore
The dimensions are 36 in by 36 in by 36 in
The volume is equal to
----> is a cube