Answer:
Original piece of metal: 23 in × 28 in
Dimensions of the box: 21 in × 26 in × 1 in
Step-by-step explanation:
Given dimensions of a rectangular piece of metal:
If squares with sides 1 in long are cut from the four corners, and the flaps are folded upwards to form an open box, 2 inches should be subtracted from the width and the length of the piece of metal. Therefore, the dimensions of the box are:
To find an expression for the volume of the box, multiply the width by the length by the height:
[tex]\begin{aligned}\implies \sf Volume &=\sf width \times length \times height\\&= (x - 2) \times (x + 3) \times 1\\& = (x-2)(x+3)\\&=x(x+3)-2(x+3)\\&=x^2+3x-2x-6\\&=x^2+x-6\end{aligned}[/tex]
If the volume is 546 in³ then:
[tex]\begin{aligned}\sf Volume&=546\\\implies x^2+x-6&=546\\x^2+x-6-546&=546-546\\x^2+x-552&=0\\x^2+24x-23x-552&=0\\x(x+24)-23(x+24)&=0\\(x-23)(x+24)&=0\\\implies x&=23,-24\end{aligned}[/tex]
As length is positive, x = 23 only.
To determine the original dimensions of the piece of metal, substitute the found value of x into the expressions for width and length. Therefore, the original dimensions of the piece of metal are:
To find the dimensions of the box, substitute the found value of x into the expressions for width and length. Therefore, the dimensions of the box are: