Question

Set up and evaluate the optimization problem. You are constructing a cardboard box from a piece of cardboard with the dimensions 4 m by 8 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

Answer 1

Answer:

$Shorter\ side=4-2\times 0.845=2.31\ m\\Longest\ side=8-2\times 0.845=6.31\ m\\Height=0.845\ m$

Step-by-step explanation:

Given that , dimension of the cardboard is 4 m by 8 m.

Lets the dimensions of the square is x m by x m.

The volume after cutting the equal size of square from all the four corners is given as

$V=x\times (4-2x)\times (8-2x)\\V=x\times (32-16x-8x+4x^2)\\V=x\times (4x^2-24x+32)\\V=4x^3-24x^2+32x$

For the maximum volume

$\dfrac{dV}{dx}=12x^2-48x+32=0\\3x^2-12x+8=0$

For maximum value of volume , the value of x will be 0.845

x= 0.845

Therefore the dimensions will be

$Shorter\ side=4-2\times 0.845=2.31\ m\\Longest\ side=8-2\times 0.845=6.31\ m\\Height=0.845\ m$