Respuesta :
Answer with Step-by-step explanation:
Let length of rectangular piece of cardboard=x
Width of rectangular piece of cardboard=y
Area of rectangular piece of cardboard=[tex]374 cm^2[/tex]
Area of rectangular piece of cardboard[tex]=l\times b=x\times y[/tex]
[tex]xy=374[/tex]
[tex]y=\frac{374}{x}[/tex]
According to question
Length of box=x-2(2)=x-4
Width of box=y-2(2)=y-4
Height of box=2
Volume of box=[tex]l\times b\times h[/tex]
Substitute the values in this formula
Volume of box=[tex]2(x-4)(y-4)[/tex]
Substitute the value of y
[tex]2(x-4)(\frac{374}{x}-4)=468[/tex]
[tex]\frac{(x-4)(374-4x)}{x}=\frac{468}{2}=234[/tex]
[tex](x-4)(374-4x)=234x[/tex]
[tex]374x-4x^2-1496+16x=234x[/tex]
[tex]4x^2-374x-16x+234x+1496=0[/tex]
[tex]4x^2-156x+1496=0[/tex]
On dividing by 4 on both sides , then we get
[tex]x^2-39x+374=0[/tex]
[tex]x^2-17x-22x+374=0[/tex]
[tex]x(x-17)-22(x-17)=0[/tex]
[tex](x-17)(x-22)=0[/tex]
[tex]x-17=0[/tex]
[tex]x=17[/tex] or [tex]x-22=0\implies x=22[/tex]
Substitute x=17
Then, we get [tex]y=\frac{374}{17}=22[/tex]
Substitute x=22
Then, we get
[tex]y=\frac{374}{22}=17[/tex]
The size of cardboard should start with 17 by 22.