Question

an open-top box is to be constructed from a sheet of tin that measures 22 inches by 14 inches by cutting out squares from each corner. let V(x) denote the volume of the resulting box. step 1 of 2: write V(x) as a product of linear factorsstep 2 of 2: among the values of x for which V(x)=0, which are physically possible?

Answer 1

It is given that an open-top box is to be constructed from a sheet of tin that measures 22 inches by 14 inches by cutting out squares from each corner

Let x be the measure of the side of the square.

Length of the resulting box =22-2x

Width of the resulting box=14-2x

Height of the resulting box=x

The volume of the box is

`V=\text{length }* width* height`

Substitute values, we get

`V(x)=(22-2x)(14-2x)x`

`=(22-2x)(14x-2x^2)`

`=22\mleft(14x-2x^2\mright)-2x\mleft(14x-2x^2\mright)`

`=22*14x-22*2x^2-2x*14x-(-2x)2x^2`

`=308x-44x^2-28x^2+4x^3`

`V(x)=4x^3-72x^2+308x`

Putting V(x)=0, we get

`4x^3-72x^2+308x=0`

`4x(x^2-18x+77)=0`

`4x=0,(x^2-18x+77)=0`

Here x is not zero

`x^2-18x+77=0`

`x^2-11x-7x+77=0`

`x(x^{}-11)-7(x-11)=0`

`(x^{}-11)(x-7)=0`

`(x^{}-11)=0\text{ or }\mleft(x-7\mright)=0`

`x^{}=11\text{ or }x=7`

The height of the box is 11 or 7

If the height is 11 inches, substitute x=11 in the length equation, we get

`\text{length =22=2x=22-2}*11=22-22=0`

we get a length is 0, so it is not possible to make the box.

Setting x=7, the height of the box is 7 inches.

`\text{Length =22-2x=22-2}*7=22-14=8inches`

`\text{width =14-2}*7=14-14=0`

we get a width is 0, so it is not possible to make the box.

Hence among the values of x for which V(x)=0 is not physically possible.