Question

15. If the scale factor of two similar solids is 3 : 10, what is the ratio of their corresponding areas and volumes?

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ratio of area 100:9; ratio of volumes 1000:27
ratio of area 9:100; ratio of volumes 27:1000
ratio of area 3:10; ratio of volume 9:100
ratio of area 10:3; ratio of volume 100:9

Answer 1

[AI-generated answer — Bot GPT-4o]

When dealing with similar solids, the relationship between the scale factor and the ratios of their corresponding areas and volumes can be determined using the following principles:

1. **Ratio of corresponding areas**: The ratio of the surface areas of two similar solids is the square of the scale factor.
2. **Ratio of corresponding volumes**: The ratio of the volumes of two similar solids is the cube of the scale factor.

Given that the scale factor of the two similar solids is 3:10, let's determine the ratios of their corresponding areas and volumes.

1. **Ratio of areas**:
The scale factor is 3:10. Therefore, the ratio of the areas is:
\[ \left(\frac{3}{10}\right)^2 = \frac{3^2}{10^2} = \frac{9}{100} \]

2. **Ratio of volumes**:
The ratio of the volumes is:
\[ \left(\frac{3}{10}\right)^3 = \frac{3^3}{10^3} = \frac{27}{1000} \]

So the correct answers are:
- Ratio of areas: \( \frac{9}{100} \)
- Ratio of volumes: \( \frac{27}{1000} \)

Hence, the correct option is:
**ratio of area 9:100; ratio of volumes 27:1000**