A spherical mirror is to be used to form on a screen 5.0 m from the object an image five times the
size of the object.
(a) Describe the type of mirror required.
(b) Where should the mirror be positioned relative to the object?

Explanation

To solve the question step by step, we will break it down into two parts: identifying the type of mirror required and determining the position of the mirror relative to the object.

### Step-by-Step Solution:

**(a) Describe the type of mirror required:**

1. **Understanding the requirements:**
- We need to form a real image that is five times the size of the object.
- A real image can only be formed by a concave mirror when the object is placed within the focal length.

2. **Conclusion:**
- Therefore, the type of mirror required is a **concave mirror**.

**(b) Where should the mirror be positioned relative to the object?**

1. **Understanding magnification:**
- The magnification (M) is given as 5, which means:
\[ M = \frac{h'}{h} = \frac{v}{u} \]
where \( h' \) is the height of the image, \( h \) is the height of the object, \( v \) is the image distance, and \( u \) is the object distance.

2. **Setting up the equation:**
- Since the image is real and inverted, we can write:
\[ M = \frac{v}{u} = 5 \]
- This implies:
\[ v = 5u \]

3. **Using the given information:**
- The distance from the object to the screen is given as 5.0 m, so:
\[ v - u = 5 \]
- Substituting \( v = 5u \) into the equation:
\[ 5u - u = 5 \]
\[ 4u = 5 \]
\[ u = 1.25 \text{ m} \]

4. **Finding the image distance:**
- Now substituting \( u \) back to find \( v \):
\[ v = 5u = 5 \times 1.25 = 6.25 \text{ m} \]

5. **Positioning the mirror:**
- The object should be placed at a distance of \( 1.25 \text{ m} \) from the mirror.

6. **Finding the focal length:**
- Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting \( v = -6.25 \text{ m} \) (real image, hence negative) and \( u = -1.25 \text{ m} \) (object distance is negative):
\[ \frac{1}{f} = \frac{1}{-6.25} + \frac{1}{-1.25} \]
\[ \frac{1}{f} = -0.16 - 0.8 = -0.96 \]
\[ f = -\frac{1}{0.96} \approx -1.04 \text{ m} \]

### Final Answers:
- **(a)** The type of mirror required is a **concave mirror**.
- **(b)** The mirror should be positioned **1.25 m away from the object**.