Final answer:
The ratio of the side lengths is 1:√7 and the ratio of the volumes is 1:343 when the ratio of the surface areas of an object is 1:7.
Step-by-step explanation:
When the ratio of the surface areas of an object that is enlarged is 1:7, it means the surface area has been scaled up by a factor of 7. The ratio of the corresponding side lengths, however, would be the square root of that scale factor because surface area scales with the square of the linear dimensions. Therefore, the ratio of the side lengths would be 1:√7.
The volume of an object scales with the cube of the linear dimensions. To find the ratio of volumes, we would cube the scale factor of the side lengths. Since we determined the side length ratio to be 1:√7, cubing this gives us 1:(√7)³, which simplifies to 1:7³ or 1:343. So the ratio of the volumes is 1:343.