A square piece of cardboard is to be used to form a box without a top by cutting off squares, 5cm on a side, from each corner and then folding up the sides. if the volume of the box must be 320 sq. sm, what must be the length of a side of the cardboard?
If the square has side x, then the box when folded up has height 5, and a square base of side x-5-5=x-10
5(x-10)^2 = 320
(x-10)^2 = 64
x-10 = 8
x = 18
Well, if we have a square piece of cardboard and we cut out squares from each corner and fold up the sides, we're essentially left with an open-top box.
Now, let's figure out the dimensions of this box. Since we cut out squares of 5cm from each corner, the remaining sides of the cardboard will be (x - 10) cm. Folding up the sides creates a box with a height of 5cm.
To find the length of a side of the cardboard, we need to determine the value of x so that the volume of the box is 320 sq. cm.
The formula for the volume of a rectangular prism is length × width × height. In this case, the length and width are both (x - 10) cm, and the height is 5cm.
Therefore, we have the equation: (x - 10) × (x - 10) × 5 = 320.
Now, let's solve this equation to find the value of x. I'm just gonna plug this into my joke calculator...
Calculating...Calculating... Oh no, I spilled coffee on my joke calculator! It's supposed to be a "brews" and advanced model, but I guess it's more of a "brew-sy" model now.
Uh, where were we? Ah yes, solving the equation.
(x - 10) × (x - 10) × 5 = 320.
*scribbles on a board*
5x^2 - 100x + 500 = 320.
5x^2 - 100x + 180 = 0.
Now we use the quadratic formula.
x = (-b ± √(b^2 - 4ac)) / 2a.
Let me do some quick math...
*more scribbles on the board*
x ≈ 6.47 cm.
So, the length of a side of the cardboard should be approximately 6.47 cm.
To find the length of a side of the cardboard, we need to work backwards from the volume of the box.
Let's assume the length of a side of the cardboard is x cm.
When we cut off squares with side length 5 cm from each corner, the resulting width and length of the base of the box will be:
Width = x - 2(5) = x - 10 cm
Length = x - 2(5) = x - 10 cm
The height of the box will be 5 cm.
To find the volume of the box, we multiply the width, length, and height:
Volume = (x - 10) * (x - 10) * 5
We are given that the volume of the box must be 320 cm^3, so we can set up the following equation:
320 = (x - 10) * (x - 10) * 5
Let's solve for x:
Divide both sides by 5:
64 = (x - 10) * (x - 10)
Take the square root of both sides:
√64 = √(x - 10) * (x - 10)
8 = x - 10
Add 10 to both sides:
8 + 10 = x - 10 + 10
18 = x
Therefore, the length of a side of the cardboard should be 18 cm.
To find the length of a side of the cardboard, we need to consider the dimensions of the resulting box after cutting off squares and folding up the sides.
Let's assume that the side length of the square piece of cardboard is "x" cm.
When we cut off 5 cm squares from each corner, the resulting box will have sides with lengths (x-10) cm. This is because we cut off 5 cm from both the length and width of the cardboard.
Now, we can calculate the volume of the box using the formula:
Volume = length × width × height
Since the box doesn't have a top, the height of the box will be equal to the size of the squares we cut off, which is 5 cm.
Given that the volume of the box is 320 cm², we have:
320 = (x-10) × (x-10) × 5
To solve this equation, let's simplify it:
320 = 5(x^2 - 20x + 100)
320 = 5x^2 - 100x + 500
5x^2 - 100x + 500 - 320 = 0
5x^2 - 100x + 180 = 0
Now, we can solve this quadratic equation to find the value(s) of "x" by factoring, completing the square, or using the quadratic formula. In this case, the equation doesn't factor nicely, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for this equation, we have:
a = 5, b = -100, c = 180
x = (-(-100) ± √((-100)^2 - 4 × 5 × 180)) / (2 × 5)
x = (100 ± √(10000 - 3600)) / 10
x = (100 ± √6400) / 10
x = (100 ± 80) / 10
Now, we have two possible values for x:
1) x = (100 + 80) / 10 = 18 cm
2) x = (100 - 80) / 10 = 2 cm
Since the length of the side cannot be 2 cm (as it's smaller than the 5 cm squares we cut off from each side), the length of the side of the cardboard must be 18 cm.