Arectangular piece of metal is 30 in longer than it is wide. squares with sides 6 in long are cut from the four corners and the flaps are folded upward to form an open box. if the volume of the box is 2706 incubed​, what were the original dimensions of the piece of​ metal?

length=53 in

width= 23 in

Step-by-step explanation:

First, we know that each corner has now a square of 6 in long, so the heigth of the box will be of this size (as you can see in the picture attached). Then, the length and the width of the box would be the same as the piece, but with 12 in less. Therefore the equations to solve the problem are:

We factorize the 6 and we get:

Finally we solve for x, and we get the initial dimensions of the piece of metal:

22 inch by 52 inch

Step-by-step explanation:

If the Width of the metal=x

The length is 30inch longer than its width= x+30

If Squares with sides 6 in long are cut from the four corners and the flaps are folded.

The length of the pre-folded box =x+30-12=(x+18)inch

The width of the pre-folded box = (x-6-6)inch = (x-12) inch

The height of the flap= 6 inch

Volume of the box =2400 inch cubed

Volume of a Cuboid=LWH

Therefore:

6(x-12)(x+18)=2400

6x²+36x-1296=2400

6x²+36x-1296-2400=0

6x²+36x-3696=0

6(x-22)(28+x)=0

x-22=0 or x+28=0

x=22 inch (Since length cannot be negative)

The original dimensions of the metal sheet are 22 inch by 52 inch

Answer = 55,25 inches

Solution -

let's take x as length and y as width of the metal piece. As per the question x is 30 more than y,

⇒ x = y + 30

Then four square pieces of side 6 are cut from each corner,

so the new length and width are

x-12 , y-12

Then the volume of the new box created will be

(x-12)(y-12)6

in the question the volume of the given figure is given to be 3354

so (x-12)(y-12)6 = 3354

putting the value of x in the the above equation

⇒ (y+30 - 12)(x-12) = 3354/6 = 559

⇒ (y+18)(y-12) = 559

⇒ y² + 6y - 775 = 0

⇒ y² + 31y - 25y -775 = 0

⇒ (y+31)(y-25) = 0

⇒ y = -31, 25

as length can not be -ve , so y = 25

then x = 25+30 = 55

Hence the dimensions of the metal piece are 55, 25 inches