To find the ratio of the volumes of sphere A to sphere B after dilation, we follow these steps:
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Calculate the radius of Sphere A: The diameter of Sphere A is 12. Therefore, the radius \( r_A \) is: \[ r_A = \frac{12}{2} = 6 \]
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Calculate the radius of Sphere B after dilation: Sphere B is created by dilating Sphere A by a scale factor of \( \frac{1}{2} \). Thus, the radius \( r_B \) of Sphere B is: \[ r_B = r_A \times \frac{1}{2} = 6 \times \frac{1}{2} = 3 \]
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Use the formula for the volume of a sphere: The volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \]
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Calculate the volume of Sphere A: Substitute \( r_A = 6 \): \[ V_A = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \times 216 = \frac{864}{3} \pi = 288 \pi \]
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Calculate the volume of Sphere B: Substitute \( r_B = 3 \): \[ V_B = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = \frac{108}{3} \pi = 36 \pi \]
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Find the ratio of the volumes: The ratio of the volume of Sphere A to Sphere B is: \[ \text{Ratio} = \frac{V_A}{V_B} = \frac{288 \pi}{36 \pi} = \frac{288}{36} = 8 \]
Thus, the ratio of the volume of Sphere A to Sphere B is: \[ \boxed{8:1} \]