1.An open box is made from a square piece of material by cutting two-cm squares
from the corners and turning up the sides. The volume of the finished box is
200𝑐𝑚3
. Find the size (area) of the original piece of materi
the sides are 2 cm high
... so the bottom of the box is a square , 10 cm on a side
the original piece was 14 cm on a side ... 10 cm + 2 cm + 2 cm
To find the area of the original square piece of material, we need to follow a step-by-step process:
Step 1: Find the dimensions of the box
Let's assume that the side length of the square piece of material is x cm. When we cut out two cm squares from each corner, the dimensions of the resulting box will be:
Length: x - 2 cm
Width: x - 2 cm
Height: 2 cm
Step 2: Calculate the volume of the box
The volume of a rectangular prism (box) is given by the formula:
Volume = Length * Width * Height
Substituting the values we found in step 1:
Volume = (x - 2) * (x - 2) * 2
Step 3: Solve for x
Since we know that the volume of the box is 200 cm^3, we can set up the equation:
200 = (x - 2) * (x - 2) * 2
Step 4: Simplify and solve the equation
Expanding and rearranging the equation:
400 = 2(x - 2)^2
200 = (x - 2)^2
±√200 = x - 2
±14.1421 = x - 2
Adding 2 to both sides:
x = 14.1421 + 2
x = 16.1421
Since the side length of the square piece of material cannot be negative, we disregard the negative solution. Therefore, x = 16.1421 cm.
Step 5: Calculate the size (area) of the original piece of material
The area of a square is given by the formula:
Area = side length * side length
Substituting the value of x:
Area = 16.1421 cm * 16.1421 cm
Area ≈ 262.14 cm^2
Therefore, the size (area) of the original piece of material is approximately 262.14 cm^2.
To find the size (area) of the original piece of material, we need to determine the dimensions of the square before the corners were cut.
Let's assume that the side length of the square piece of material is "x" cm.
When two cm squares are cut from each corner, the resulting box will have a height of 2 cm (since the squares are turned up to form the sides of the box).
The length and width of the base of the box will be reduced by 4 cm (2 cm on each side) due to the corners being cut. Therefore, the dimensions of the base will be (x - 4) cm.
The volume of a box is given by the formula V = length * width * height.
In this case, the volume of the box is 200 cm^3 and the height is 2 cm. We can substitute these values into the formula:
200 = (x - 4) * (x - 4) * 2
Simplifying the equation:
100 = (x - 4) * (x - 4)
Expanding the equation:
100 = x^2 - 8x + 16
Rearranging the equation:
x^2 - 8x + 16 - 100 = 0
x^2 - 8x - 84 = 0
We can now solve this quadratic equation for x using factoring, completing the square, or the quadratic formula.
Using the quadratic formula: x = (-(-8) ± √((-8)^2 - 4(1)(-84))) / (2(1))
Simplifying the equation:
x = (8 ± √(64 + 336)) / 2
x = (8 ± √400) / 2
x = (8 ± 20) / 2
This gives us two possible solutions for x:
x = (8 + 20) / 2 = 28 / 2 = 14
x = (8 - 20) / 2 = -12 / 2 = -6
Since length cannot be negative in this context, we discard the negative solution.
Therefore, the side length of the original square piece of material is 14 cm.
The size (area) of the original piece of material is given by the formula area = side length * side length:
Area = 14 cm * 14 cm
Area = 196 cm^2
So, the size (area) of the original piece of material is 196 cm^2.