## You are constructing a cardboard box from a piece of cardboard with the dimensions 2 m by 4 m. You then cut equal-size squares from each cor

Question

You are constructing a cardboard box from a piece of cardboard with the dimensions 2 m by 4 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions (in m) of the box with the largest volume

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Math
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2021-09-13T22:49:07+00:00
2021-09-13T22:49:07+00:00 1 Answer
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## Answers ( )

Answer:Dimensions:

l =3,15 m

w=1,15 m

x= 0,42 m (height)

V(max) = 1,52 m³

Step-by-step explanation:The cardboard is L = 4 m and W = 2 mLet call x the length of the square to cut in each corner, then, volume of open box is:For the side L is L – 2*x l = 4 – 2*x

For the side W is W – 2*x w= 2 – 2*x

The height is x

Volume of the open box, as function of x is:

V(x) = ( 4 -2x) * ( 2 – 2x) *x ⇒ V(x) = ( 8 – 8x -4x + 4x²) *x

V(x) = ( 8 -12x + 4x² )*x V(x) = 8x – 12x² + 4x³

V(x) = 8x – 12x² + 4x³

Taking derivatives on both sides of the equation

V´(x) = 8 – 24x + 12x²

V´(x) = 0 8 – 24x + 12x² = 0 reordering 12x² – 24x + 8 = 0

or 3x² – 6x + 2 = 0

A second degree equation. Solving for x

x₁,₂ = ( 6 ± √36 – 24 ) /6

x₁,₂ = ( 6 ± 3.46) /6

x₁ = 6 + 3,46 /6 x₁ = 1.58 we dismiss such solution because 1,58 * 2 = 3,15 and is bigger than 2 one of the side of the cardboard

x₂ =( 6 – 3,46 ) / 6

x₂ = 0,42 m

Dimensions of the open box

l = 4 – 2*x l = 4 – 0,85 l = 3,15 m

w = 2 -2*x w = 2 – 0,85 w = 1,15 m

x = 0,42 m

V(max) =3,15*1,15*0,42

V(max) = 1,52 m³