A convex mirror has an object placed 15 cm in front of it. The image is formed 5 cm behind the mirror. Calculate the focal length of the mirror.

For a convex mirror:
The object is placed in front of the mirror, so the object distance $u$ is negative.

$$u = -15 \text{ cm}$$

The image formed by a convex mirror is always virtual and formed behind the mirror, so the image distance $v$ is positive.

$$v = +5 \text{ cm}$$

The mirror formula is given by:

$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

where $f$ is the focal length.

$$\frac{1}{f} = \frac{1}{5 \text{ cm}} + \frac{1}{-15 \text{ cm}}$$ $$\frac{1}{f} = \frac{1}{5 \text{ cm}} - \frac{1}{15 \text{ cm}}$$

To subtract the fractions, find a common denominator, which is 15.

$$\frac{1}{f} = \frac{3}{15 \text{ cm}} - \frac{1}{15 \text{ cm}}$$ $$\frac{1}{f} = \frac{3 - 1}{15 \text{ cm}}$$ $$\frac{1}{f} = \frac{2}{15 \text{ cm}}$$

Now, invert both sides to find $f$:

$$f = \frac{15 \text{ cm}}{2}$$ $$f = 7.5 \text{ cm}$$

The focal length of the convex mirror is $*7.5 \text{ cm}*$.