Question

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 17 in. long and 16 in. wide, find the dimensions of the box that will yield the maximum volume.

Answer 1

Answer:

The maximum volume is: 11.52 *10.52*2.74  = 332.06 $in^{3}$

Step-by-step explanation:

Let x is the side of the square in inches.(x >0)

The volume of the resulting box when the flaps are folded up can be expressed as:

V  = x (17 -2x)(16-2x)

= (17x -2$x^{2}$)(16-2x)

= $4x^{3}$ - 66$x^{2}$ +  272x

To find the value of x which yields the maximum of volume (V), take the first derivative of this equation and set it equal to zero.

$\frac{dV}{dx}$ = 12$x^{2}$ - 66x + 272

Solve 12$x^{2}$ - 132x + 272  = 0

<=> x = 2.74

The dimensions of the box are:

Length: (17 -2x) = (17 -2*2.74) = 11.52

Width: (16-2x) = (16-2*2.74) = 10.52

High: x = 2.74

So the maximum volume is: 11.52 *10.52*2.74  = 332.06 $in^{3}$