A concave mirror has a radius of curvature of 20 cm. If an object is placed 30 cm away from the mirror, where is the image formed?

Step 1: Identify given values and determine the focal length.
The object distance is $u = 30 \text{ cm}$.
The radius of curvature of the concave mirror is $R = 20 \text{ cm}$.

For a concave mirror, the focal length $f$ is half the radius of curvature and is negative according to the Cartesian sign convention.

$$f = -\frac{R}{2}$$ $$f = -\frac{20 \text{ cm}}{2}$$ $$f = -10 \text{ cm}$$

According to the Cartesian sign convention, for a real object placed in front of the mirror, the object distance $u$ is negative.

$$u = -30 \text{ cm}$$

Step 2: Calculate the image position ($v$) using the mirror formula.
The mirror formula is:

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

Substitute the values of $f$ and $u$:

$$\frac{1}{-10 \text{ cm}} = \frac{1}{v} + \frac{1}{-30 \text{ cm}}$$

Rearrange the formula to solve for $v$:

$$\frac{1}{v} = \frac{1}{-10 \text{ cm}} - \frac{1}{-30 \text{ cm}}$$ $$\frac{1}{v} = -\frac{1}{10 \text{ cm}} + \frac{1}{30 \text{ cm}}$$

To combine the fractions, find a common denominator, which is 30:

$$\frac{1}{v} = \frac{-3}{30 \text{ cm}} + \frac{1}{30 \text{ cm}}$$ $$\frac{1}{v} = \frac{-3 + 1}{30 \text{ cm}}$$ $$\frac{1}{v} = \frac{-2}{30 \text{ cm}}$$ $$\frac{1}{v} = -\frac{1}{15 \text{ cm}}$$ $$v = -15 \text{ cm}$$