To solve this problem involving a concave mirror, let’s analyze step by step:

Given:

1. Object height ($h_o$) = $35.0 \, \text{cm}$
2. Object distance ($u$) = $30 \, \text{cm}$ (in front of the mirror → taken as negative for concave mirrors)
3. Focal length ($f$) = $15.0 \, \text{cm}$ (also negative since it’s a concave mirror)

Key Mirror Formula:

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

• $f$ = focal length
• $v$ = image distance
• $u$ = object distance

Rearranging for $v$ (image distance): $\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$

Step 1: Calculate Image Distance ($v$)

Substitute $f = -15 \, \text{cm}$ and $u = -30 \, \text{cm}$: $\frac{1}{v} = \frac{1}{-15} - \frac{1}{-30}$ Simplifying: $\frac{1}{v} = -\frac{1}{15} + \frac{1}{30}$ $\frac{1}{v} = -\frac{2}{30} + \frac{1}{30}$ $\frac{1}{v} = -\frac{1}{30}$ Taking the reciprocal: $v = -30 \, \text{cm}$

Thus, the image distance is $30 \, \text{cm}$ (negative indicates it’s on the same side as the object).

Step 2: Calculate Magnification ($m$)

The magnification formula is: $m = -\frac{v}{u}$ Substitute $v = -30 \, \text{cm}$ and $u = -30 \, \text{cm}$: $m = -\frac{-30}{-30} = -1$

So, the magnification is $-1$.

Step 3: Calculate Image Height ($h_i$)

The image height can be calculated using: $h_i = m \cdot h_o$ Substitute $m = -1$ and $h_o = 35.0 \, \text{cm}$: $h_i = -1 \cdot 35.0 = -35.0 \, \text{cm}$

The image height is $35.0 \, \text{cm}$ (negative indicates the image is inverted).

Step 4: Describe the Image

• Type of image: Real
• Size of the image: Same size as the object
• Orientation of the image: Inverted

Final Answers:

1. Magnitude of the image distance: $30 \, \text{cm}$
2. Magnitude of the image height: $35.0 \, \text{cm}$
3. Magnitude of the magnification: $1$
4. Type of image formed: Real
5. Size of the image: Same size
6. Orientation of the image: Inverted

Questions for Further Practice:

1. What happens to the image if the object is placed at the focal point?
2. How does the image change if the object distance is doubled?
3. Derive the mirror formula mathematically.
4. What are the characteristics of an image formed by a convex mirror?
5. What type of image is formed if the magnification is $+0.5$?

Tip: Always use the correct sign convention (negative for concave mirrors) while solving problems involving spherical mirrors. Let me know if you want further clarifications!