Calculate the focal length of a concave spherical mirror that projects an image 12.0 cm in front of the mirror with a magnitude of -0.42

To calculate the focal length of a concave spherical mirror, we'll use the mirror formula:

1/f = 1/v - 1/u

where:
- f is the focal length,
- v is the image distance (distance of the image from the mirror), and
- u is the object distance (distance of the object from the mirror).

In this case, we are given:
- v = -12.0 cm (as the image is projected in front of the mirror, the distance is taken as negative),
- magnification (m) = -0.42 (the negative sign indicates the image is inverted).

To find the focal length, we need to determine the object distance (u) first. Rearranging the magnification formula, we get:

m = -v/u

Substituting the given values:

-0.42 = -12.0/u

Now we can solve for u:

12.0/u = 0.42

u = 12.0 / 0.42

u ≈ 28.57 cm

Now, we can substitute the values of u and v into the mirror formula to find the focal length:

1/f = 1/-12.0 - 1/28.57

Calculating this equation gives us:

1/f = -0.0833 - 0.035

1/f ≈ -0.1183

Finally, we can find f by taking the reciprocal of both sides:

f = 1 / -0.1183

f ≈ -8.46 cm

Therefore, the focal length of the concave spherical mirror is approximately -8.46 cm.