Final Answer:
The correct answer is d. 27.
Step-by-step explanation:
When a sphere is dilated by a factor of \(k\), the ratio of the volumes of the original sphere (\(V_{\text{original}}\)) to the dilated sphere (\(V_{\text{dilated}}\)) is given by \(k^3\). In this case, the sphere is dilated by a factor of 3. Therefore, the ratio of the volumes is \(3^3 = 27\). This means that the volume of the new sphere is 27 times the volume of the original sphere.
To elaborate further, the formula for the volume of a sphere is \(V = \frac{4}{3}\pi r^3\), where \(r\) is the radius. When dilating the sphere by a factor of 3, the new radius becomes \(3r\). Substituting this into the volume formula for both the original and dilated spheres and calculating the ratio \(k^3\), we obtain \(3^3 = 27\). This indicates that the volume of the dilated sphere is 27 times the volume of the original sphere.
In conclusion, the correct ratio of volumes from the original sphere to the new sphere, when dilated by a factor of 3, is 27. This demonstrates the cubic relationship between the scale factor and the volume of a sphere in dilation, highlighting the expansion in three dimensions.