A company is selling 1 gallon blocks of orange juice. your task is to design a rectangle box ( prism) that meet all the criteria below:

Question

a) contains at least 231 cubic inches.
b)the height of the box must be exactly 1/4th of the width of the box.
c)the box is closed on all 6 sides.
d) the surface area of the box minimized.?

Answer 1

let the height be x
then the width is 4x , (the height is 1/4 of the width, this way I have no fractions )
let the length be y
volume = (x)(4x)(y) = 4x^2 y
4x^2 y = 231
y = 231/(4x^2)

SA = 2(4xy) +2(x)(4x) + 2(xy)
= 8xy + 8x^2 + 2xy
= 10xy+8x^2
= 10x(231/x^2) + 8x^2
= 2310/x + 8x^2

d(SA)/dx = -2310/x^2 + 16x
= 0 for a min of SA
16x = 2310/x^2
x^2 = 2310/16 = 1155/8
x = (1155)^(1/3)/2 = appr 5.25 inches
4x = 21.0 inches
y = 2.10 inches

The box should be 5.25 inches by 21 inches by 2.1 inches

check:
for my answer:
5.25 x 21 x 2.1 = 231.55
SA = 660.5

if x = 5
Volume = 231
SA = 662 which is > 660.5

if x = 4
volume = 231
SA = 705.5 which is > 660.5

My answer is correct

Answer 2

Hmm. I got

v = 5.29 x 13.22 x 3.30 = 230.78
a = 262.12

Answer 3

yup, looked over my work and found an error in
my SA substitution.

SA = 2(4xy) +2(x)(4x) + 2(xy)
....
= 10x(231/x^2) + 8x^2 ---->10x(231/(4x^2)) + 8x^2

to produce your answers