A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 5 cm squares from each corner and folding up the sides. Let x represent the width of the original piece of cardboard.
Determine a function V that represents the volume of the box in terms of x.
A) $V ( x ) = 10 x ^ { 2 } - 75 x + 125$
B) $V ( x ) = 10 x ^ { 2 }$
C) $V ( x ) = 5 x ^ { 2 } - 100 x + 500$
D) $V ( x ) = 10 x ^ { 2 } - 150 x + 500$

Question

A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 5 cm squares from each corner and folding up the sides. Let x represent the width of the original piece of cardboard.
Determine a function V that represents the volume of the box in terms of x. 
A) $V ( x ) = 10 x ^ { 2 } - 75 x + 125$
B) $V ( x ) = 10 x ^ { 2 }$
C) $V ( x ) = 5 x ^ { 2 } - 100 x + 500$
D) $V ( x ) = 10 x ^ { 2 } - 150 x + 500$

Quizplus · Accepted Answer

The volume of the box, $V$, after cutting out the squares and folding, can be expressed as $V = lwh$, where $l$ is the length, $w$ is the width, and $h$ is the height of the box. Given that the original piece of cardboard has a width of $x$ and a length twice that, or $2x$, and that 5 cm squares are cut from each corner, the new dimensions of the box will be:- Length: $2x - 2(5) = 2x - 10$- Width: $x - 2(5) = x - 10$- Height: $5$ cm (the size of the squares cut out)Therefore, the volume function in terms of $x$ is:$$V(x) = (2x - 10)(x - 10)(5) = 10(x^2 - 10x + 10x - 100) = 10(x^2 - 100) = 10x^2 - 1000$$However, there seems to be a mistake in my calculation as none of the options match exactly. The correct approach to find the volume after cutting and folding is indeed to adjust the length and width by subtracting twice the cutout size (since cuts are made on both sides) and to use the cutout size as the height. The correct formula, accounting for the error in my calculation, should reflect the adjustments made to both the length and width due to the cutouts and the multiplication by the height, which is correctly represented in option D:$$V(x) = (2x - 10)(x - 10)(5)$$This simplifies correctly to:$$V(x) = 10x^2 - 150x + 500$$This accounts for the decrease in both length and width by 10 cm (5 cm from each side) and multiplies by the height (5 cm) to find the volume.

A Piece of Cardboard Is Twice as Long as It V(x)=10x2−75x+125V ( x ) = 10 x ^ { 2 } - 75 x + 125V(x)=10x2−75x+125

A Piece of Cardboard Is Twice as Long as It $V ( x ) = 10 x ^ { 2 } - 75 x + 125$