Question

A rectangular cardboard sheet has a length that is 1.5 times greater than the width. Is it possible to make a topless box with a volume of 6080 cm3 out of this cardboard sheet if squares with a side of 8 cm are cut from the corners? Find the dimensions of the cardboard sheet.

Answer 1

54 cm length and 36 cm width.

Explanation:

Let's call
`x` the width of the cardboard sheet,
`1.5x` is the length of it (the problem states that the length is 1.5 times the width).

Also,
`y=8cm` which is the removed part of the box.

Remember that the volume of a rectangular prism is the product between each dimension. In this case, we have

`V=(1.5x-2y)(x-2y)(y)=6080`

Where we already included the removed part of the box.

Replacing values, we have

`(1.5x-2(8))(x-2(8))(8)=6080\\(1.5x-16)(8x-128)=6080\\12x^(2) -192x-128x+2048=6080\\12x^(2)-320x+2048-6080=0\\12x^(2)-320x-4032=0`

Using a calculator, we have

`x_(1)=36\\x_(2) \approx -9.33`

Where only the positive number make sense to the problem, because there's no negative lengths.

So, the length is
`1.5x=1.5(36)=54`

Therefore, the dimensions are 54 cm length and 36 cm width.

Answer 2

Yes, it is possible to make a topless box with a volume of 6080 cm3 out of this cardboard sheet.

The dimensions of the cardboard sheet are 54 cm x 36 cm

Explanation:

Let

x ----> the length of the cardboard sheet

y ----> the width of the cardboard sheet

we know that

`x=1.5y` ----> equation A

The volume of the topless box is

`V=LWH`

where

`V=6,080\ cm^3`

`L=x-2(8)=(x-16)\ cm`

`W=y-2(8)=(y-16)\ cm`

`H=8\ cm`

substitute

`6,080=(x-16)(y-16)8` ----> equation B

substitute equation A in equation B

`6,080=(1.5y-16)(y-16)8`

`6,080/8=(1.5y-16)(y-16)`

`760=1.5y^2-24y-16y+256`

`760=1.5y^2-40y+256`

`1.5y^2-40y+256-760=0`

`1.5y^2-40y-504=0`

Solve for y

Solve the quadratic equation by graphing

The solution is y=36 cm

see the attached figure

Find the value of x

`x=1.5y` ---->
`x=1.5(36)=54\ cm`

therefore

Yes, it is possible to make a topless box with a volume of 6080 cm3 out of this cardboard sheet.

The dimensions of the cardboard sheet are 54 cm x 36 cm