Question

Jay wants to make a box, with no lid (or top), out of a 10" x 6" rectangular piece of cardboard. If Jay cuts squares with dimensions x by x out of each corner of the cardboard, and then folds up the corners to make an open box, find a function that represents: (a) The volume of the box. (b) The surface area of the box.

Answer 1

a.
`V(x)=x(10-2x)(6-2x)`

b.
`S(x)=4(15-x^2)`

Explanation:

Dimensions of rectangular piece of cardboard=
`10* 6`

According to question

Length of box,l=10-2x

Breadth of box,b=6-2x

Height of box,h=x

a.

Volume of box=
`l* b* h`

Substitute the values in the formula

Volume of box=
`x(10-2x)(6-2x)`

`V(x)=x(10-2x)(6-2x)`

b.Surface area of box=
`2(b+l)h+lb`

Because the box has no lid

Substitute the values in the formula

Surface area of box=
`2(10-2x+6-2x)x+(10-2x)(6-2x)`

Surface area of box=
`2(16-4x)x+60-20x-12x+4x^2`

Surface area of box=
`32x-8x^2+60-32x+4x^2`

Surface area of box=
`60-4x^2=4(15-x^2)`

Surface area of box, S(x)=
`4(15-x^2)`