A small statue has a height of 3.8 cm and is placed in front of a concave mirror. The image of the statue is inverted, 1.9 cm tall, and located 14 cm away from the mirror. Find the focal length of the mirror.

To find the focal length of the concave mirror, we can make use of the mirror equation, which connects the focal length (f), the object distance (do), and the image distance (di). The equation is as follows:

1/f = 1/do + 1/di

Given information:
Object height (ho) = 3.8 cm
Image height (hi) = -1.9 cm (negative sign indicates the inversion)
Image distance (di) = -14 cm (negative sign indicates the position behind the mirror)

First, we need to find the object distance (do) using the magnification formula:

magnification (m) = hi/ho = -di/do

Substituting the given values:

-1.9 cm / 3.8 cm = -14 cm / do

Now, we can solve for the object distance (do):

do = (-14 cm * 3.8 cm) / -1.9 cm = 28 cm

Now we have the object distance (do) and the image distance (di). We can substitute these values into the mirror equation to find the focal length (f):

1/f = 1/do + 1/di

1/f = 1/28 cm + 1/(-14 cm)

1/f = (1 - 2) / 28 cm

1/f = -1 / 28 cm

Now, taking the reciprocal of both sides of the equation, we find:

f = -28 cm

Therefore, the focal length of the concave mirror is -28 cm.