Final answer:
The distance between the mirror and the image created by the convex mirror is approximately 8.54 cm. This negative distance indicates that the image is virtual and located behind the mirror.
Step-by-step explanation:
To find the distance between the mirror and the image created by a convex mirror, we can use the mirror equation 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance. For a convex mirror, the focal length is negative, so we will use -11.7 cm in our calculation. With the object placed 31.7 cm from the mirror, we can rearrange the mirror equation to solve for di:
1/(-11.7) = 1/31.7 + 1/di
1/di = 1/(-11.7) - 1/31.7
1/di = -0.08547008547 - 0.03154574132
1/di = -0.1170158268
di = -1/0.1170158268
di ≈ -8.54 cm
The negative sign indicates that the image is virtual and located behind the mirror. The distance between the mirror and the image is therefore approximately 8.54 cm.