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
[tex]x=1.5y[/tex] ----> equation A
The volume of the topless box is
[tex]V=LWH[/tex]
where
[tex]V=6,080\ cm^3[/tex]
[tex]L=x-2(8)=(x-16)\ cm[/tex]
[tex]W=y-2(8)=(y-16)\ cm[/tex]
[tex]H=8\ cm[/tex]
substitute
[tex]6,080=(x-16)(y-16)8[/tex] ----> equation B
substitute equation A in equation B
[tex]6,080=(1.5y-16)(y-16)8[/tex]
[tex]6,080/8=(1.5y-16)(y-16)[/tex]
[tex]760=1.5y^2-24y-16y+256[/tex]
[tex]760=1.5y^2-40y+256[/tex]
[tex]1.5y^2-40y+256-760=0[/tex]
[tex]1.5y^2-40y-504=0[/tex]
Solve for y
Solve the quadratic equation by graphing
The solution is y=36 cm
see the attached figure
Find the value of x
[tex]x=1.5y[/tex] ----> [tex]x=1.5(36)=54\ cm[/tex]
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:
Step-by-step explanation:
Let's call [tex]x[/tex] the width of the cardboard sheet, [tex]1.5x[/tex] is the length of it (the problem states that the length is 1.5 times the width).
Also, [tex]y=8cm[/tex] 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
[tex]V=(1.5x-2y)(x-2y)(y)=6080[/tex]
Where we already included the removed part of the box.
Replacing values, we have
[tex](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[/tex]
Using a calculator, we have
[tex]x_{1}=36\\x_{2} \approx -9.33[/tex]
Where only the positive number make sense to the problem, because there's no negative lengths.
So, the length is [tex]1.5x=1.5(36)=54[/tex]
Therefore, the dimensions are 54 cm length and 36 cm width.
