From a rectangular piece of cardboard having dimensions a × b, where a = 40 inches and b = 70 inches, an open box is to be made by cutting out an identical square of area x2 from each corner and turning up the sides (see the figure). Express the volume V of the box in terms of x. (Factor your answer completely.)
Well, I'm no mathematician, but let's give this a shot!
First, let's figure out the dimensions of the box after we cut out the squares. Since the squares have an area of x^2, their sides will each be x inches long. That means the length and width of the box will be (40 - 2x) and (70 - 2x) inches, respectively.
Now, to find the volume of the box, we multiply the length, width, and height. Since the height is x inches, the volume V can be expressed as:
V = (40 - 2x)(70 - 2x)(x)
To make it easier to work with, let's factor it completely:
V = x(40 - 2x)(70 - 2x)
So, the volume V of the box is x multiplied by the expression (40 - 2x)(70 - 2x). I hope that helps! And remember, if all else fails, you can always use the box to store your collection of clown noses!
To find the volume of the box, we need to determine the dimensions of the box after the square corners are cut out and the sides are folded up.
The length of the box will be the original length of the cardboard minus twice the length of each side of the square cut out. Since each side of the square cut out has length x, the length of the box will be (40 - 2x) inches.
Similarly, the width of the box will be (70 - 2x) inches.
The height of the box will be the length of the square cut out, which is x inches.
Therefore, the volume V of the box can be expressed as:
V = length × width × height
V = (40 - 2x) × (70 - 2x) × x
To factor this expression, we can expand it first and then simplify:
V = (40 - 2x) × (70 - 2x) × x
= (40 × 70 × x) - (2x × 70) - (2x × 40) + (4x^2)
= 2800x - 140x - 80x + 4x^2
= 4x^2 - 220x + 2800
Therefore, the volume of the box, V, in terms of x is:
V = 4x^2 - 220x + 2800
To express the volume V of the box in terms of x, we need to find the dimensions of the open box first and then calculate its volume.
Let's denote the length of each side of the square cut out from each corner as "x". As shown in the figure, after cutting out the squares, the dimensions of the remaining piece of the cardboard will be (a - 2x) and (b - 2x).
Therefore, the length, width, and height of the open box will be (a - 2x), (b - 2x), and x, respectively.
The volume V of a rectangular box is given by the product of its length, width, and height. So the volume of the open box can be calculated as:
V = (a - 2x) * (b - 2x) * x
Substituting the values of a and b:
V = (40 - 2x) * (70 - 2x) * x
To simplify this expression, we can expand it:
V = (40*70*x - 2x*70 - 2x*40 + 4x^2) * x
V = (2800x - 140x^2 - 80x + 4x^2) * x
Now, let's combine the like terms:
V = (2800x - 140x^2 - 80x + 4x^2) * x
V = (2800x - 180x^2) * x
Finally, let's factor out x completely:
V = x * (2800 - 180x)
Therefore, the volume V of the box in terms of x is given by V = x(2800 - 180x).
after turning up the corners, the bottom now measures
(a-2x)(b-2x)
The depth of the box is the size of the tabs folded up, or x
V = (a-2x)(b-2x)(x)
= x(40-2x)(70-2x)
= 4x(20-x)(35-x)