SOLUTION: A box with an open top is to be made by taking a rectangular piece of tin 8 by 10 inches and cutting a square of the same size out of each corner and folding up the sides. If the a

It can be solved in several different ways.


        For example, a  mental solution  is as follows.


The original rectangular piece of tin has dimensions 8 by 10 inches.

The difference between the two dimensions is 10-8 = 2 inches.

After cutting a square of the same size out of each corner and folding up the sides,
the difference between the dimensions of the base will be the same 2 inches.

So, you need to find the dimensions of the rectangular base, given that the area is 
24 square inches, while the difference of the dimensions is 2 inches.


At this point a person, familiar with the multiplication table, will guess the answer 
in 5-6 seconds: the dimensions of the base are 6 by 4 inches.


So, the side of the squares to cut are  (10-6)/2 = 4/2 = 2 inches.


And the problem is just solved mentally to the end this way.



        Algebraic solution  uses equations.


It starts by noticing that if x is the size of the square to cut from each corner,
then, after folding up the sides, the dimensions of the base are (10-2x) inches by (8-2x) inches.


It leads to the equation for the base area

    (10-2x)*(8-2x) = 24  square inches.


So, we should find x from this equation.


Simplify it step by step, but as the first step, divide by sides by 2*2 = 4 to get

    (5-x)*(4-x) = 6,

    20 - 4x - 5x + x^2 = 6

    x^2 - 9x + 14 = 0

    (x-7)*(x-2) = 0     <<<---===  factoring.


Thus two possible roots to the equation are  x= 2  and x= 7.


Bit 7 inches square side is too much to cut (obviously).


It leaves only one remaining possibility x= 2 inches, which is the correct ANSWER.