Question

Four congruent square pieces with lengths of sides of 6 cm were cut out of corners of a square piece of cardboard. Then this piece of cardboard was folded into an open-top box. Find the original dimensions of the piece of cardboard if the volume of the resulting box is 486 cm^3.

Answer 1

The original dimensions of the piece of cardboard, from which congruent squares were cut to create an open-top box with a volume of 486 cm³, are 21 cm by 21 cm.

Step-by-step explanation:

The student's question involves finding the original dimensions of a piece of cardboard, from which four congruent squares with sides of 6 cm were cut and then folded to form an open-top box with a volume of 486 cm³.

To solve this problem, we'll denote the side of the original square as x cm. Because four squares of side 6 cm are removed from the corners, the dimensions of the box, when created, will be (x - 2*6) cm by (x - 2*6) cm by 6 cm. The volume of this box can be calculated as:

V = l × w × h

V = (x - 12) × (x - 12) × 6

Given that the volume is 486 cm³, we can set up the equation:

486 = (x - 12)^2 × 6

Upon solving this equation for x, we get:

81 = (x - 12)^2

x - 12 = 9 (since x > 12 for the cardboard to be large enough)

x = 21 cm

Thus, the original dimensions of the piece of cardboard are 21 cm × 21 cm.

Answer 2

The original dimensions of the cardboard is 21 cm × 21 cm, if a square pieces with lengths of sides of 6 cm were cut out of corners of a square piece of cardboard and the volume of the resulting box is 486 cubic centimeters.

Step-by-step explanation:

The given is,

Square pieces with lengths of sides of 6 cm were cut out of corners of a square piece of cardboard.

The volume of the resulting box is 486 cubic centimeters.

Step:1

The volume, is the given piece of cardboard was folded into an open box, we calculate side value of cardboard after cut into a square piece,

Formula for volume of cardboard is,

`Volume, V = whl`......................(1)

Where, w - width

h - Height

l - length

From given, V= 486 cubic centimeters

h = 6 cm ( Ref attachment)

w = l

Equation (1) becomes,

`486=6(l^(2) )`

`l^(2) = 81`

Take root on both sides

l = 9 centimeters

Step:2

Ref the attachment,

Side value of piece of cardboard

= 6 + 9 + 6

= 21

Side value of piece of cardboard = 21 centimeters

original sides of cardboard is 21 cm × 21 cm

Result:

The original dimensions of the cardboard is 21 cm × 21 cm, if a square pieces with lengths of sides of 6 cm were cut out of corners of a square piece of cardboard and the volume of the resulting box is 486 cubic centimeters.