Question

A 3.00 cm tall object is held 20.00 cm away from a concave mirror whose radius of curvature is 30.00 cm. What is the image height?
Group of answer choices

The image is 4 times as far from the mirror as the object.

The image is 3 times as far from the mirror as the object.

The image is the same distance as far from the mirror as the object.

The image is 2 times as far from the mirror as the object.

Answer 1

To solve this problem, we will use the mirror formula and magnification formula.

### Step 1: Find the focal length (f) of the concave mirror.
The focal length \( f \) is half the radius of curvature \( R \):
\[
f = \frac{R}{2} = \frac{30 \, \text{cm}}{2} = 15 \, \text{cm}
\]

### Step 2: Use the mirror formula to find the image distance (q).
The mirror formula is given by:
\[
\frac{1}{f} = \frac{1}{p} + \frac{1}{q}
\]
Where:
- \( f \) is the focal length
- \( p \) is the object distance
- \( q \) is the image distance

Given that the object distance \( p = 20 \, \text{cm} \):
\[
\frac{1}{15} = \frac{1}{20} + \frac{1}{q}
\]

Now, rearranging the equation to isolate \( \frac{1}{q} \):
\[
\frac{1}{q} = \frac{1}{15} - \frac{1}{20}
\]

Finding a common denominator (60):
\[
\frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}
\]

Substituting these values in:
\[
\frac{1}{q} = \frac{4}{60} - \frac{3}{60} = \frac{1}{60}
\]

Thus,
\[
q = 60 \, \text{cm}
\]

### Step 3: Calculate the magnification (m).
The magnification formula is:
\[
m = -\frac{q}{p} = \frac{h'}{h}
\]

Where:
- \( h' \) is the image height
- \( h \) is the object height

From our earlier calculations:
\[
m = -\frac{60}{20} = -3
\]

### Step 4: Find the image height.
Using the magnification to find the image height:
\[
m = \frac{h'}{h} \quad \Rightarrow \quad -3 = \frac{h'}{3.00 \, \text{cm}}
\]
\[
h' = -3 \times 3.00 \, \text{cm} = -9.00 \, \text{cm}
\]

The negative sign indicates that the image is inverted.

### Conclusion:
The image height is \( 9.00 \, \text{cm} \) (inverted).

### How far is the image from the mirror compared to the object?
The image distance \( q \) (60 cm) is 3 times the object distance \( p \) (20 cm).

So the correct statement is: **The image is 3 times as far from the mirror as the object.**