Answer:
4 in × 4 in × 8 in or Â
6.47 in × 6.47 in × 3.06 in
Step-by-step explanation:
Data:
(1)          V = 128 in³
(2) Â Â Â Â Â Â Â Â Â l = w = x
(3) 4(l + w + h) = 64 in  (There are 12 edges)
Calculation:
The formula for the volume of the box is
(4) Â Â Â Â Â Â Â Â V = lwh
(5)        128 = x²h      Substituted (1) and (2) into (4)
(6)          h = 128/x²   Divided each side by x²
     l + w +h = 16       Divided (1) by 4
     x + x + h = 16      Substituted (2) into 6
(7) Â Â Â Â Â 2x + h = 16 Â Â Â Â Â Combined like terms
   2x + 128/x² = 16      Substituted (6) into (7)
    2x³ + 128 = 16x²     Multiplied each side by x²
2x³ - 16x²+ 128 = 0       Subtracted 16x² from each side
  x³ - 8x² + 64 = 0      Divided each side by 2
According to the Rational Zeros theorem, a rational root must be a positive or negative factor of 64.
The possible factors are ±1, ±2, ±4, ±8, ±16, ±32, ± 64.
After a little trial-and-error with synthetic division (start in the middle and work down) we find that x = 4 is a zero. Â Â Â Â Â Â Â
4|1 Â -8 Â Â 0 Â 64
 |    4  -16  -64
  1  -4  -16    0
So, the cubic equation factors into (x - 4)(x² - 4x + 16) = 0
We can use the quadratic formula to find that the roots of the quadratic are
x = 2 - 2√5 and x = 2+ 2√5
We reject the negative value and find that there are two solutions to the problem.
x = 4 in and x = 2 + 2√5 ≈ 6.472 in
Case 1. x = 4 in
h = 128/x² = 128/4² = 128/16 = 8 in
The dimensions of the box are 4 in × 4 in × 8 in
Also, 4(l + w + h) = 4( 4 + 4 + 8) = 4 × 16 =  64 in
Case 2. x = 6.472 in
h = 128/x² = 128/6.472² = 128/41.89 = 3.056 in
The dimensions of the box are 6.47 in × 6.47 in × 3.06 in
Also, 4(l + w + h) = 4( 6.47 + 6.47 + 3.06) = 4 × 16.00 =  64 in
The two solutions are
(a) 4 in    × 4 in    × 8 in
(b) 6.47 in × 6.47 in × 3.06 in  Â
