A convex mirror with a radius of curvature of 34.0 cm forms a 0.95 cm tall image of a pencil at a distance of 12.5 cm behind the mirror. Calculate the object distance for the pencil and its height. Is the image real or virtual? What is the magnification?

To solve this problem, we can use the mirror formula and magnification formula.

The mirror formula is given by:

1/f = 1/u + 1/v

where:
f is the focal length of the mirror,
u is the object distance,
v is the image distance.

We know that the radius of curvature (R) of the convex mirror is 34.0 cm. The focal length (f) of a convex mirror is half the radius of curvature, so:

f = R/2 = 34.0/2 = 17.0 cm

Given that the image distance (v) is 12.5 cm behind the mirror, we can substitute these values into the mirror formula to find the object distance (u):

1/17.0 = 1/u + 1/12.5

Solving this equation will give us the value of u.

Now, let's calculate the height of the object using the magnification formula:

magnification (m) = -v/u = h'/h

Given that the height of the image (h') is 0.95 cm, we can substitute this value along with the calculated value of u.

Finally, we can determine whether the image is real or virtual by analyzing the sign of the magnification. If the magnification is positive, the image is virtual. If the magnification is negative, the image is real.

So, to summarize:
1. Use the mirror formula and the value of v to calculate the value of u.
2. Use the magnification formula and the values of v, u, and h' to calculate the height of the object.
3. Analyze the sign of the magnification to determine if the image is real or virtual.