A box with a lid is to be cut out of a 12 inch by 24 inch sheet of thin cardboard by cutting out six x-inch squares and folding them. What are the dimensions to two decimal places of all possible boxes that will have a volume of 100 cubic inches?
To find the dimensions of the box that will have a volume of 100 cubic inches, we need to set up an equation using the given information.
Let's assume that the length of the side of the square cut out is "x". Since we have to cut out six squares, the length of the box (excluding the flaps) will be (12 - 2x) inches, and the width of the box will be (24 - 2x) inches.
The height of the box is unknown at this point, so we'll denote it as "h".
The formula for the volume of a rectangular box is given by V = length * width * height. In this case, V = 100 cubic inches.
Therefore, we can set up the equation:
(12 - 2x) * (24 - 2x) * h = 100
Simplifying the equation, we get:
(288 - 72x - 48x + 4x^2) * h = 100
(4x^2 - 120x + 288) * h = 100
4x^2 - 120x + 288 = 100 / h
Now, we have an equation in terms of x and h. We can solve this equation to find the possible values of x.
To get the dimensions to two decimal places, we can substitute different values for h and solve for x using numerical methods, such as iterative methods or graphing calculators.
For example, let's assume h = 1 and solve the equation:
4x^2 - 120x + 288 = 100 / 1
4x^2 - 120x + 188 = 0
Using numerical methods, we find that one possible solution for x is approximately 3.92.
Similarly, we can repeat this process for different values of h to find other possible values of x.
Once we have the values of x, we can substitute them back into the expressions for the length and width of the box to find their dimensions.
Note: Since the equation is quadratic, it is possible that there may be no real solutions for x or that there are multiple solutions. Therefore, it's important to use numerical methods to find the precise values.