Afsana is using a cake recipe where the cake should be baked in a pan of 12 inches diameter. If she uses half the amount of the recipe ingredients to make a cake of the same depth, by what factor should she change the diameter of the pan?

To determine the factor by which Afsana should change the diameter of the pan when using half the recipe ingredients, we need to compare the volumes of the original cake and the half-quantity cake. Since the depth of the cake remains the same, the volume of the cake will be proportional to the area of the base, which is a circle.

### Step-by-step solution:

1. **Original Cake Volume**:
   The area of a circular base is given by \( A = \pi r^2 \), where \( r \) is the radius. For a pan with a 12-inch diameter, the radius is:
   \[
   r = \frac{12}{2} = 6 \text{ inches}
   \]
   So, the area of the original pan is:
   \[
   A_{\text{original}} = \pi \times 6^2 = 36\pi \text{ square inches}
   \]

2. **Half Recipe Volume**:
   Since Afsana is using half the amount of ingredients, the volume of the new cake will be half of the original cake's volume. With the depth of the cake unchanged, the area of the new pan must also be halved. Therefore, the area of the new pan should be:
   \[
   A_{\text{new}} = \frac{1}{2} \times A_{\text{original}} = \frac{1}{2} \times 36\pi = 18\pi \text{ square inches}
   \]

3. **New Radius**:
   Let \( r_{\text{new}} \) be the radius of the new pan. The area of the new pan is:
   \[
   A_{\text{new}} = \pi \times r_{\text{new}}^2
   \]
   Set this equal to the calculated area of the new pan:
   \[
   \pi \times r_{\text{new}}^2 = 18\pi
   \]
   Dividing both sides by \( \pi \), we get:
   \[
   r_{\text{new}}^2 = 18
   \]
   Taking the square root:
   \[
   r_{\text{new}} = \sqrt{18} = 3\sqrt{2} \approx 4.24 \text{ inches}
   \]

4. **New Diameter**:
   The new diameter is:
   \[
   \text{New diameter} = 2 \times r_{\text{new}} = 2 \times 4.24 \approx 8.48 \text{ inches}
   \]

5. **Factor of Change**:
   The factor by which Afsana should change the diameter is the ratio of the new diameter to the original diameter:
   \[
   \text{Factor} = \frac{8.48}{12} \approx 0.707
   \]

### Conclusion:
Afsana should reduce the diameter of the pan by a factor of approximately \( 0.707 \) or about \( 70.7\% \) of the original diameter.

Would you like more details on any part of the solution?

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**Relative Questions:**
1. How does the volume of the cake relate to the area of the base when the depth remains constant?
2. If Afsana used a quarter of the ingredients, by what factor should she change the diameter?
3. What happens to the volume of the cake if both the depth and diameter are halved?
4. Can you find the new diameter for any other fractional changes in the recipe?
5. How would the factor change if the pan shape were rectangular instead of circular?

**Tip**: When adjusting dimensions for similar shapes, areas scale with the square of the linear dimension, while volumes scale with the cube.

Learn how to adjust the diameter of a cake pan when using half the recipe ingredients while keeping the same cake depth. This problem explores geometry concepts, such as the area of a circle and proportional scaling. Suitable for students in Grades 9-11.

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