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

Answer:

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

Step-by-step 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

Answer 2

Answer:

54 cm length and 36 cm width.

Step-by-step 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.