Final answer:
To determine the dimensions of the original sheet metal, one must denote the width as W inches and setup an equation considering the cuts and folds to make the tray, using the given volume of 1620 cubic inches to solve for W and consequently find the sheet's length.
Step-by-step explanation:
The student is asking to find the dimensions of the original piece of sheet metal that, when the corners are cut off and flaps folded to make an open tray, results in a volume of 1620 cubic inches. The tray's volume is the product of its length, width, and height. To start, let's denote the width of the original metal sheet as W inches and thus its length as W - 7 inches, since the length is 7 inches shorter than the width.
After cutting out the 3x3 inch squares from each corner, the new dimensions of the tray will be (W - 3 - 3) inches in width and (W - 7 - 3 - 3) inches in length, with a height of 3 inches (the size of the squares cut out). So, the volume V of the tray is given by:
V = (W - 6) * (W - 13) * 3 cubic inches
Set up an equation using the given volume:
1620 = (W - 6) * (W - 13) * 3
Now solve the quadratic equation for W. The positive solution will give the width of the rectangular metal sheet. The length can then be determined by subtracting 7 inches from W. The resulting dimensions are the answer to the student's question.