To solve these problems, we can use the formulas for the circumference of a circle and relationships between circles.
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Scale Factor between Circle X and Circle Z:
The circumference of a circle \( C \) is given by the formula: \[ C = 2\pi r \] Where \( r \) is the radius of the circle.
If the circumference of circle X is 12 m, we can calculate the radius of circle X as follows: \[ C_X = 2\pi r_X \implies 12 = 2\pi r_X \implies r_X = \frac{12}{2\pi} = \frac{6}{\pi} \]
If the circumference of circle Z is 6 m, we can calculate the radius of circle Z: \[ C_Z = 2\pi r_Z \implies 6 = 2\pi r_Z \implies r_Z = \frac{6}{2\pi} = \frac{3}{\pi} \]
Now we can find the scale factor (SF) between the two circles using their radii: \[ SF = \frac{r_X}{r_Z} = \frac{\frac{6}{\pi}}{\frac{3}{\pi}} = \frac{6}{3} = 2 \] This means the scale factor from circle Z to circle X is 2:1, which corresponds to option \( 01:2 \) (assuming the order is the other way around).
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Circumference of Circle O' after Dilation:
The circumference of circle O is calculated as: \[ C_O = 2\pi r_O \] Given that the radius \( r_O \) is 14 meters: \[ C_O = 2\pi(14) = 28\pi \]
If circle O is dilated by a scale factor of 2.5, the new radius \( r_{O'} \) will be: \[ r_{O'} = 2.5 \times 14 = 35 \text{ meters} \]
Now we can calculate the circumference of circle O': \[ C_{O'} = 2\pi r_{O'} = 2\pi(35) = 70\pi \]
To find the approximate circumference in meters, we can use \( \pi \approx 3.14 \): \[ C_{O'} \approx 70 \times 3.14 = 219.8 \text{ m} \]
So, rounding this gives us approximately 220 m.
Therefore, the answers are:
- Scale Factor: 1:2
- Circumference of Circle O': 220 m
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