by cutting a square out of each corner of a sheet of card (30cm by 21cm)and folding up the sides,i ca make an open box with a capacity of 1080cm^3.What is the area in cm^2 of each of the squares i have to cut out?
this problem has been solved in many of the related questions below, with different numbers. Follow some of them and give it a shot. What do you get?
Let the sides you cut out be x centimetres long.
The volume is then:
(30-2x)(21-2x)x = 1080
Expanding:
4 x^3 - 102 x^2 + 630 x - 1080 = 0
Look for common factors:
(x-3)(4x^2 - 90x + 360) = 0
Found one root, solve the quadratic for the other.
To find the area of each square that needs to be cut out, we'll need to follow these steps:
Step 1: Determine the dimensions of the box
When we cut out squares from each corner of the card and fold up the sides, the resulting box will have a length, width, and height. We can determine these dimensions based on the original dimensions of the card and the volume of the box.
Given:
Length of the card = 30 cm
Width of the card = 21 cm
Volume of the box = 1080 cm³
To find the height of the box, we can rearrange the formula for volume:
Volume = length × width × height
Plugging in the values:
1080 = 30 × 21 × height
Simplifying the equation, we find:
height = 1080 / (30 × 21)
height = 2 cm
So, the dimensions of the box are:
Length = original length of the card - 2 * length of square
Width = original width of the card - 2 * width of square
Height = 2 cm
Step 2: Calculate the side length of the square
Since we removed squares from each corner, and the height of the box is 2 cm, we can calculate the dimensions of each square.
For the length:
Length = original length of the card - 2 * length of square
30 = 30 - 2 * side length of square
2 * side length of square = 0
side length of square = 0
Similarly, for the width:
Width = original width of the card - 2 * width of square
21 = 21 - 2 * side length of square
2 * side length of square = 0
side length of square = 0
Step 3: Determine the area of each square
Since the side length of each square is 0 cm, the area of each square is also 0 cm².
Therefore, the area of each square that needs to be cut out is 0 cm².