Final answer:
To find the scale factor to fit two standard-sized papers onto a legal-sized paper, divide the area of the legal paper (119 sq in) by the total area of two standard papers (187 sq in), resulting in a scale factor of approximately 0.636.
Step-by-step explanation:
The question is asking to find the scale factor k needed to resize a standard piece of paper so that two of them can fit onto a legal-sized piece of paper, without changing the aspect ratio. To find the scale factor, we should compare the areas of standard and legal-sized papers as we want to fit two standard papers on one legal-sized paper.
The area of the standard paper (8.5 inches by 11 inches) is 93.5 square inches. The area of the legal-sized paper (8.5 inches by 14 inches) is 119 square inches. Two standard-sized papers would have a total area of 187 square inches.
As we cannot really fit two full-sized standard papers onto a legal-sized one, we need the combined area of two dilated standard papers to be equal or less than the area of the legal-sized paper. Therefore, the scale factor k is found by dividing the area of the legal-sized paper by the combined area of two standard papers, which is k = 119 / 187.
Calculate the scale factor:
- Area of standard paper = 8.5 * 11 = 93.5 sq in
- Area of 2 standard papers = 93.5 * 2 = 187 sq in
- Area of legal paper = 8.5 * 14 = 119 sq in
- Scale factor k = 119 / 187
Hence the required scale factor k is approximately 0.636 when rounded to three decimal places.