A rectangular piece of​ cardboard, whose area is 374 square​ centimeters, is made into an open box by cutting a 2​-centimeter square from each corner and turning up the sides. If the box is to have a volume of 468 cubic​ centimeters, what size cardboard should you start​ with?

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.