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 distance?
Group of answer choices

1.5 cm

15 cm

30 cm

60 cm

Answer 1

To find the image distance for the concave mirror, we can use the mirror formula:

\[
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
\]

where:
- \( f \) is the focal length of the mirror
- \( d_o \) is the object distance (positive in front of the mirror)
- \( d_i \) is the image distance (positive if the image is real and on the same side as the object)

### Step 1: Calculate the focal length (f)

The focal length \( f \) of a concave mirror is related to its radius of curvature \( R \) by the formula:

\[
f = \frac{R}{2}
\]

Given that the radius of curvature \( R = 30.00 \, cm \):

\[
f = \frac{30.00 \, cm}{2} = 15.00 \, cm
\]

### Step 2: Use the mirror formula

Now we have:
- \( f = 15.00 \, cm \) (positive for concave mirror)
- \( d_o = 20.00 \, cm \) (positive value)

Substituting these values into the mirror formula:

\[
\frac{1}{15.00} = \frac{1}{20.00} + \frac{1}{d_i}
\]

### Step 3: Solve for \( \frac{1}{d_i} \)

First, calculate \( \frac{1}{20.00} \):

\[
\frac{1}{20.00} = 0.05
\]

Now substitute into the mirror formula:

\[
\frac{1}{15.00} = 0.05 + \frac{1}{d_i}
\]

Calculating \( \frac{1}{15.00} \):

\[
\frac{1}{15.00} \approx 0.06667
\]

Now set up the equation:

\[
0.06667 = 0.05 + \frac{1}{d_i}
\]

Subtract \( 0.05 \) from both sides:

\[
0.06667 - 0.05 = \frac{1}{d_i}
\]

This gives:

\[
0.01667 = \frac{1}{d_i}
\]

### Step 4: Solve for \( d_i \)

Taking the reciprocal to find \( d_i \):

\[
d_i = \frac{1}{0.01667} \approx 60.00 \, cm
\]

### Conclusion

Therefore, the image distance \( d_i \) is approximately \( 60.00 \, cm \). From the given choices, the correct answer is:

**60 cm**