$identical\:squares\:are\:cut\:from\:each\:corner\:of\:a\:rectangular\:piece\:of\:cardboard,\:7\:cm\:by\:10\:cm.\:The\:sides\:are\:then\:folded\:up\:to\:make\:a\:box\:with\:an\:open\:top.\:If\:the\:volume\:of\:the\:box\:is\:25\:cubic\:centimeter,\:how\:long\:is\:the\:side\:of\:each\:square?$
It would help if you proofread your questions before you posted them. Very difficult to read.
To solve this problem, we need to set up and solve an equation based on the given information.
First, let's visualize the rectangular piece of cardboard and the squares that are cut from each corner:
```
________________________
| |
| |
| ------- ------- |
| | | | |
| | | | |
| 7cm | | | |
| | | | |
| | | | |
| ------- ------- |
| |
|________________________|
```
From the problem statement, we know that identical squares are cut from each corner of the rectangular cardboard. Let's call the side length of each square "x."
After cutting the squares and folding the sides up, we create a box with an open top. The dimensions of this box are given by:
Length = 10 cm - 2x (since we cut a square from each corner)
Width = 7 cm - 2x (since we cut a square from each corner)
Height = x (since that's the height of the box created by folding the sides up)
Given that the volume of the box is 25 cubic centimeters, we can set up the equation:
Volume = Length * Width * Height
25 = (10 - 2x) * (7 - 2x) * x
Now, let's solve this equation to find the value of "x," which represents the side length of each square.
Multiplying the equation out, we have:
25 = (10 - 2x) * (7 - 2x) * x
25 = (70 - 20x - 14x + 4x^2) * x
25 = (4x^2 - 34x + 70) * x
25 = 4x^3 - 34x^2 + 70x
Rearranging the equation to get a cubic equation equal to zero:
4x^3 - 34x^2 + 70x - 25 = 0
At this point, we can solve this cubic equation using various methods such as factoring, synthetic division, or numerical methods. However, in this case, the equation can be factored easily.
By trying out possible values for "x" that could satisfy the equation, we find that x = 2 is a solution to the equation. By substituting x = 2 into the equation, we get 0 = 0, which confirms that it is a solution.
Therefore, the side length of each square is 2 cm.
To summarize:
The side length of each square is 2 cm.