Alright, let's solve this step by step.
Step 1: Understand the initial dimensions of the map and the paper.
- The map has dimensions 2 feet (width) by 3 feet (height).
- The paper has dimensions 8 inches (width) by 10 inches (height).
Step 2: Convert the paper dimensions from inches to feet.
- There are 12 inches in a foot.
- Convert the width of the paper: [tex]\( \frac{8 \text{ inches}}{12} = \frac{8}{12} = \frac{2}{3} \)[/tex] feet.
- Convert the height of the paper: [tex]\( \frac{10 \text{ inches}}{12} = \frac{10}{12} = \frac{5}{6} \)[/tex] feet.
So, the paper dimensions are [tex]\( \frac{2}{3} \)[/tex] feet (width) by [tex]\( \frac{5}{6} \)[/tex] feet (height).
Step 3: Calculate the scale factors for width and height.
- Determine the scale factor needed to fit the map width into the paper width: [tex]\( \text{scale\_width} = \frac{\frac{2}{3} \text{ feet} (paper)}{2 \text{ feet} (map)} \)[/tex].
- Determine the scale factor needed to fit the map height into the paper height: [tex]\( \text{scale\_height} = \frac{\frac{5}{6} \text{ feet} (paper)}{3 \text{ feet} (map)} \)[/tex].
So:
[tex]\[ \text{scale\_width} = \frac{2}{3} \div 2 = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} \][/tex]
[tex]\[ \text{scale\_height} = \frac{5}{6} \div 3 = \frac{5}{6} \times \frac{1}{3} = \frac{5}{18} \][/tex]
Step 4: Determine the smaller of the two scale factors to maintain the aspect ratio.
- Compare [tex]\( \text{scale\_width} \)[/tex] and [tex]\( \text{scale\_height} \)[/tex]: [tex]\(\frac{1}{3} \)[/tex] and [tex]\( \frac{5}{18} \)[/tex].
- The smaller scale factor is [tex]\( \frac{5}{18} \)[/tex].
Step 5: Use the smaller scale factor to calculate the largest possible dimensions of the map that can fit on the page.
- Calculate the final width of the map: [tex]\( \text{final\_map\_width} = 2 \text{ feet} \times \frac{5}{18} = \frac{10}{18} = \frac{5}{9} \)[/tex] feet.
- Calculate the final height of the map: [tex]\( \text{final\_map\_height} = 3 \text{ feet} \times \frac{5}{18} = \frac{15}{18} = \frac{5}{6} \)[/tex] feet.
So, the dimensions of the largest possible map that can fit on the page are [tex]\(\frac{5}{9}\)[/tex] feet (approximately 0.5556 feet) by [tex]\(\frac{5}{6}\)[/tex] feet (approximately 0.8333 feet).
Thus, the final dimensions for the largest possible map that can fit on the paper are roughly:
[tex]\( 0.5556 \)[/tex] feet by [tex]\( 0.8333 \)[/tex] feet.
