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We are given that a photograph is originally **4 ½ cm** long and **2 ⅔ cm** high. It is enlarged without distortion, meaning the proportion between the length and height is maintained. The new length of the enlarged photograph is **6 cm**, and we need to determine the new height.

### Step-by-Step Solution:

1. **Find the original ratio of the length to height**:
   - Original length = \( 4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{9}{2} \) cm
   - Original height = \( 2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{8}{3} \) cm
   - Ratio of original length to original height:
   \[
   \frac{\text{length}}{\text{height}} = \frac{\frac{9}{2}}{\frac{8}{3}} = \frac{9}{2} \times \frac{3}{8} = \frac{27}{16}
   \]

2. **Set up the proportion for the new height**:
   Let the new height be \( h \). The ratio of the new length to the new height should remain the same as the original ratio. Thus:
   \[
   \frac{6}{h} = \frac{27}{16}
   \]
   Cross-multiply to solve for \( h \):
   \[
   6 \times 16 = 27 \times h
   \]
   \[
   96 = 27h
   \]
   \[
   h = \frac{96}{27} = \frac{32}{9} \approx 3 \frac{5}{9} \, \text{cm}
   \]

### Conclusion:
The new height of the enlarged photograph is \( \mathbf{3 \frac{5}{9}} \) cm, which corresponds to option **B**.

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Would you like more details or have any questions about this? Here are 5 related questions to explore further:

1. How would you calculate the height if the photograph was enlarged to a different length?
2. What would happen to the proportions if the photograph were distorted during enlargement?
3. Can you calculate the area of the original and enlarged photographs?
4. What if the photograph was reduced instead of enlarged? How would you find the new height?
5. How do ratios help maintain the aspect ratio in enlargements?

**Tip**: When enlarging or reducing shapes without distortion, always maintain the ratio between dimensions to avoid changing their appearance.

A photograph is 4 1/2 cm long and 2 2/3 cm high. It is enlarged without distortion so that the new length is 6 cm. What is the new height?

Learn how to calculate the new height of a photograph when enlarged without distortion, maintaining the original aspect ratio. The photo originally measured 4 1/2 cm by 2 2/3 cm and was enlarged to 6 cm in length.

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