An open-topped box is constructed from a piece of cardboard with a length 2 cm longer

than its width. A 6 cm square is cut from each corner and the flaps turned up from the sides
of the box. If the volume of the box is 4050 cm3 , find the dimensions of the original piece of
the cardboard.

original size: w(w+2)

after folding, the box is

(w-12)(w+2-12)6 = 4050
w = 37

so, the original piece was 37x39

1443 cm

To solve this problem, we can break it down into a series of steps:

Step 1: Understanding the problem:
The problem describes an open-topped box that is made from a piece of cardboard. The cardboard has a length that is 2 cm longer than its width. Additionally, a square measuring 6 cm is cut from each corner, and the resulting flaps are turned up to form the sides of the box. The problem asks us to find the dimensions of the original piece of cardboard, given that the volume of the box is 4050 cm^3.

Step 2: Defining variables:
Let's define the width of the cardboard as "x" cm. Since the length is 2 cm longer, the length would be "x + 2" cm.

Step 3: Constructing the box:
By cutting squares of side 6 cm from each corner, the height of the box is 6 cm. Since we are left with flaps from the remaining sides, the width and length of the base of the box would be "x - 12" cm and "x + 2 - 12" cm, respectively.

Step 4: Calculating the volume of the box:
The volume of a rectangular box can be calculated by multiplying its length, width, and height. In this case, the volume of the box is given as 4050 cm^3. So, we can set up the following equation:

Volume of the box = Length × Width × Height
4050 cm^3 = (x + 2 - 12) cm × (x - 12) cm × 6 cm

Step 5: Solving the equation:
Now, we can solve the equation by simplifying and rearranging it:

4050 cm^3 = (x - 10)(x - 12)(6)
4050 cm^3 = 6(x - 10)(x - 12)
675 cm^3 = (x - 10)(x - 12)

Step 6: Solving the quadratic equation:
To solve the quadratic equation, we can expand and rearrange it:

0 = x^2 - 22x + 120
0 = (x - 12)(x - 10)

Step 7: Finding the values of x:
From the equation, we get two potential values for x: x = 12 and x = 10.

Step 8: Checking the feasible solution:
Since the width cannot be smaller than the length, x = 10 is not a feasible solution. Therefore, x = 12 is the correct width of the cardboard.

Step 9: Calculating the length:
The length of the cardboard is given as x + 2, so the length would be 12 + 2 = 14 cm.

Step 10: Final answer:
The dimensions of the original piece of cardboard are 12 cm by 14 cm.