You're correct in observing a sign convention issue in the calculation. Here’s a breakdown of why:

In mirror formula calculations for concave mirrors, the convention is as follows:

• Focal length $f$ is positive for a concave mirror (as given here).
• The object distance $d_o$ is positive (since it’s on the same side as the incoming light).
• The image distance $d_i$ will be negative if the image forms on the same side as the object (real image) and positive if it’s on the opposite side (virtual image).

Let's go through the calculation with the proper signs:

Given:

• $f = +10$ cm (positive for a concave mirror).
• $d_o = 30$ cm.

Using the mirror equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Rearrange to solve for $\frac{1}{d_i}$: $\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{10} - \frac{1}{30}$

Calculate the right side: $\frac{1}{10} - \frac{1}{30} = \frac{3 - 1}{30} = \frac{2}{30} = \frac{1}{15}$

Thus: $d_i = +15 \, \text{cm}$

So, the answer is indeed $+15$ cm, indicating a real image formed on the same side as the object. The positive sign shows the image is real, which is typical for a concave mirror with the object outside the focal point.

The initial solution is correct as per the sign convention applied. Your observation about $-1/15$ would apply if the object were within the focal length, leading to a virtual image (which would yield a negative image distance).

Would you like further details on mirror conventions or have any additional questions?

Here are five questions that extend this topic:

1. What would the image distance be if the object were placed within 10 cm of the mirror?
2. How would the sign of $d_i$ change for a convex mirror in a similar scenario?
3. What is the nature of the image when an object is placed exactly at the focal point of a concave mirror?
4. How does the image distance vary with changes in object distance for concave mirrors?
5. What are the practical applications of concave mirrors in everyday optics?

Tip: Always double-check the sign conventions based on the type of mirror and the placement of the object to avoid calculation errors.