Sure! Let's go through the process to find the dimensions of the largest possible map that can fit on the piece of paper, step-by-step:
### Step 1: Convert Map Dimensions to Inches
Given the map dimensions:
- Map width: 2 feet
- Map height: 3 feet
We'll convert these feet measurements to inches (since there are 12 inches in a foot):
- Map width in inches: [tex]\( 2 \text{ ft} \times 12 \text{ in/ft} = 24 \text{ in} \)[/tex]
- Map height in inches: [tex]\( 3 \text{ ft} \times 12 \text{ in/ft} = 36 \text{ in} \)[/tex]
### Step 2: Calculate Scaling Factors
The dimensions of the piece of paper are:
- Paper width: 8 inches
- Paper height: 10 inches
Next, we calculate the scaling factors to determine how much the map needs to be reduced to fit within the paper:
- Width scaling factor: [tex]\( \frac{\text{paper width}}{\text{map width}} = \frac{8 \text{ in}}{24 \text{ in}} = \frac{1}{3} \)[/tex]
- Height scaling factor: [tex]\( \frac{\text{paper height}}{\text{map height}} = \frac{10 \text{ in}}{36 \text{ in}} \approx 0.2778 \)[/tex]
### Step 3: Choosing the Smallest Scaling Factor
To ensure the map fits within the paper while maintaining its aspect ratio, we use the smallest scaling factor. Therefore, the smallest factor is approximately 0.2778 (or [tex]\( \frac{10}{36} \)[/tex]).
### Step 4: Calculate Scaled Map Dimensions
Now, we determine the dimensions of the largest possible map that can fit on the paper by applying this smallest scaling factor:
- Scaled map width: [tex]\( \text{map width} \times \text{smallest scale factor} = 24 \text{ in} \times 0.2778 \approx 6.6667 \text{ in} \)[/tex]
- Scaled map height: [tex]\( \text{map height} \times \text{smallest scale factor} = 36 \text{ in} \times 0.2778 = 10 \text{ in} \)[/tex]
### Step 5: Final Dimensions
Therefore, the dimensions of the largest possible map that can fit on the piece of paper are approximately 6.6667 inches by 10 inches.
In mixed fraction form, 6.6667 inches is equivalent to [tex]\( 6 \frac{2}{3} \)[/tex] inches.
Thus, the correct dimensions are:
[tex]\( 6 \frac{2}{3} \)[/tex] in. by 10 in.
Hence, the correct answer choice is:
[tex]$6 \frac{2}{3}$[/tex] in. by 10 in.
