Question

A cardboard box without a lid is to be made with a volume of 44 ft3. Find the dimensions of the box that requires the least amount of cardboard.

Answer 1

Answer:

x =  3.53 ft

y - 3.53 ft

z = 3.53 ft

Step-by-step explanation:

given details

volume = 44 ft^3

let cardboard dimension is x and y and height be z

we know that area of given cardboard without lid is given as

A = xy + 2xy + 2yz

xyz   = 44 ft^3

To minimize area we have

A = xy + 2x (44/xy) + 2y(44/xy)

A = xy + (44/y) + (44/x)

we have

$Ax = y - \frac{44}{x^2}$

$0 = yx^2 = 44$................1

$Ay = x - \frac{44}{y^2}$

$0 = x - \frac{44}{y^2}$

$xy^2 = 44$ ..............2

from 1 and 2

$yx^2 = xy^2$

xy(y-x) = 0

xy = 0 or y = x

from geometry of probelem

x ≠ 0 and y ≠ 0

so y = x

x^3 = 44

x =  3.53 ft = y

z = 44/xy = 3.53