Question

0. Folded boxes a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Squares with sides of length x are cut out of each corner of a square piece of cardboard with sides of length /. Find the volume of the largest open box that can be formed in this way.

Answer 1

Box dimensions:

L = 3 ft

D = 6 ft

H = 1 ft

Volume V = 18 ft³

Explanation:

a) Each side of the future box is

L = 5 - 2*x D = 8 - 2*x

height of the box is x

Then the volume of the box as function of x is:

V(x) = ( 5 - 2*x ) * ( 8 - 2*x ) * x

V(x) = ( 40 - 10*x - 16*x + 4*x² ) * x

V(x) = 40*x - 26*x² + 4*x³

Tacking derivatives on both sides of the equation:

V´(x) = 40 - 52*x + 12x² or V´(x) = 20 - 26*x + 6*x²

If V´(x) = 0

6*x² - 26*x + 20 = 0

Solving for x

x₁,₂ = (-b ± √ b² - 4*a*c ] /2*a

x₁,₂ = ( 26 ± √ 676 - 480 ) / 12

x₁,₂ = (26 ± 14 )/ 12

x₁ = 1

x₂ = 3,33 ( we dismiss this solution since is not feasible according to cardboard dimensions )

Then the sides of the box are:

L = 5 -2*x L = 5 - 2*1 L = 3 ft

D = 8 - 2*x D = 8 - 2*1 D = 6 ft

H = x = 1 ft

And the volume is V = 6*3*1 = 18 ft³