To determine which dimensions could be used to create a scale model of the garden that measures [tex]\(4 \, \text{m} \times 20 \, \text{m}\)[/tex], we need to check if the given options maintain the same scale ratio as the original garden.
Let's examine each option:
### Original Garden Dimensions:
- Length: [tex]\( 20 \, \text{m} \)[/tex]
- Width: [tex]\( 4 \, \text{m} \)[/tex]
### Option A: [tex]\( 40 \, \text{cm} \times 200 \, \text{cm} \)[/tex]
First, convert the given dimensions from centimeters to meters:
- Length: [tex]\( 200 \, \text{cm} = 2.0 \, \text{m} \)[/tex]
- Width: [tex]\( 40 \, \text{cm} = 0.4 \, \text{m} \)[/tex]
Now calculate the ratios:
- Length ratio: [tex]\( \frac{2.0 \, \text{m}}{20 \, \text{m}} = 0.1 \)[/tex]
- Width ratio: [tex]\( \frac{0.4 \, \text{m}}{4 \, \text{m}} = 0.1 \)[/tex]
Since both ratios are equal, the dimensions (40 cm × 200 cm) maintain the same scale as the original garden.
### Option B: [tex]\( 1 \, \text{cm} \times 5 \, \text{m} \)[/tex]
First, convert the dimensions if necessary:
- Width: [tex]\( 1 \, \text{cm} = 0.01 \, \text{m} \)[/tex]
- Length is already in meters: [tex]\( 5 \, \text{m} \)[/tex]
Now calculate the ratios:
- Length ratio: [tex]\( \frac{5 \, \text{m}}{20 \, \text{m}} = 0.25 \)[/tex]
- Width ratio: [tex]\( \frac{0.01 \, \text{m}}{4 \, \text{m}} = 0.0025 \)[/tex]
The ratios are not equal, so these dimensions do not maintain the same scale as the original garden.
### Option C: [tex]\( 2 \, \text{m} \times 4 \, \text{m} \)[/tex]
- Length: [tex]\( 2 \, \text{m} \)[/tex]
- Width: [tex]\( 4 \, \text{m} \)[/tex]
Now calculate the ratios:
- Length ratio: [tex]\( \frac{2 \, \text{m}}{20 \, \text{m}} = 0.1 \)[/tex]
- Width ratio: [tex]\( \frac{4 \, \text{m}}{4 \, \text{m}} = 1.0 \)[/tex]
The ratios are not equal, so these dimensions do not maintain the same scale as the original garden.
### Option D: [tex]\( 1 \, \text{m} \times 4 \, \text{m} \)[/tex]
- Length: [tex]\( 4 \, \text{m} \)[/tex]
- Width: [tex]\( 1 \, \text{m} \)[/tex]
Now calculate the ratios:
- Length ratio: [tex]\( \frac{4 \, \text{m}}{20 \, \text{m}} = 0.2 \)[/tex]
- Width ratio: [tex]\( \frac{1 \, \text{m}}{4 \, \text{m}} = 0.25 \)[/tex]
The ratios are not equal, so these dimensions do not maintain the same scale as the original garden.
### Conclusion:
After examining all the options, the dimensions that maintain the same scale as the original garden [tex]\(4 \, \text{m} \times 20 \, \text{m}\)[/tex] are:
[tex]\[ \boxed{40 \, \text{cm} \times 200 \, \text{cm}} \][/tex]
