Question

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?

Answer 1

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=$374 cm^2$

Area of rectangular piece of cardboard$=l\times b=x\times y$

$xy=374$

$y=\frac{374}{x}$

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=$l\times b\times h$

Substitute the values in this formula

Volume of box=$2(x-4)(y-4)$

Substitute the value of y

$2(x-4)(\frac{374}{x}-4)=468$

$\frac{(x-4)(374-4x)}{x}=\frac{468}{2}=234$

$(x-4)(374-4x)=234x$

$374x-4x^2-1496+16x=234x$

$4x^2-374x-16x+234x+1496=0$

$4x^2-156x+1496=0$

On dividing by 4 on both sides , then we get

$x^2-39x+374=0$

$x^2-17x-22x+374=0$

$x(x-17)-22(x-17)=0$

$(x-17)(x-22)=0$

$x-17=0$

$x=17$ or $x-22=0\implies x=22$

Substitute x=17

Then, we get $y=\frac{374}{17}=22$

Substitute x=22

Then, we get

$y=\frac{374}{22}=17$

The size of cardboard should  start with  17 by 22.