To solve this problem, let's start by determining the dimensions of the sheet metal. Let's say the width of the sheet metal is "x" inches. Since the length is 2.6 times the width, the length of the sheet metal will be 2.6x inches.
To make a box with an open top, we need to cut 3-inch squares from each corner of the sheet metal. This will reduce the width and length of the sheet metal by 6 inches (3 inches on each side).
After cutting the squares and folding up the sides, the dimensions of the box will be as follows:
Width: x - 6 inches
Length: 2.6x - 6 inches
Height: 3 inches
To find the volume of the box, we multiply the width, length, and height:
Volume = (x - 6) * (2.6x - 6) * 3
Now, we need to find the range of values for x that will produce a volume between 600 and 800 cubic inches.
Let's set up the inequalities based on this requirement:
600 <= (x - 6) * (2.6x - 6) * 3 <= 800
We can solve this inequality to find the values of x that satisfy the volume requirement. To simplify the process, let's divide the entire inequality by 3:
200 <= (x - 6) * (2.6x - 6) <= 266.67
Now, let's solve the inequalities separately:
For the lower bound:
200 <= (x - 6) * (2.6x - 6)
Expand the equation:
200 <= 2.6x^2 - 6x - 15.6x + 36
Combine like terms:
200 <= 2.6x^2 - 21.6x + 36
Rearrange the equation to form a quadratic inequality:
2.6x^2 - 21.6x + 36 - 200 >= 0
2.6x^2 - 21.6x - 164 >= 0
Now, we can solve this quadratic inequality using factoring, quadratic formula, or graphing methods. Once we find the solution, we can write it as x >= [solution].
For the upper bound:
(x - 6) * (2.6x - 6) <= 266.67
Expand the equation:
2.6x^2 - 6x - 15.6x + 36 <= 266.67
Combine like terms:
2.6x^2 - 21.6x + 36 <= 266.67
Rearrange the equation to form a quadratic inequality:
2.6x^2 - 21.6x + 36 - 266.67 <= 0
2.6x^2 - 21.6x - 230.67 <= 0
Again, we can solve this quadratic inequality using factoring, quadratic formula, or graphing methods. Once we find the solution, we can write it as x <= [solution].
By solving the individual inequalities, we can find the range of values for x that will produce the desired range of volumes for the box.