To find the volume of the smaller object, we can use the relationship between the volumes of similar objects, which is proportional to the cube of the ratio of their corresponding sides.
Given:
- Corresponding side lengths: 4 cm (smaller) and 7 cm (larger)
- Volume of the larger object: 230 cm³
First, we calculate the ratio of the sides:
\[ \text{Ratio} = \frac{4}{7} \]
Next, we find the ratio of the volumes, which is the cube of the ratio of the sides:
\[ \left(\frac{\text{Volume of smaller object}}{\text{Volume of larger object}}\right) = \left(\frac{4}{7}\right)^3 \]
Calculating \(\left(\frac{4}{7}\right)^3\):
\[ \left(\frac{4}{7}\right)^3 = \frac{64}{343} \]
Now, setting up the equation for the volumes:
\[ \frac{V_{small}}{230} = \frac{64}{343} \]
To find \(V_{small}\), we rearrange the equation:
\[ V_{small} = 230 \times \frac{64}{343} \]
Calculating \(V_{small}\):
\[ V_{small} \approx 230 \times 0.1860 \approx 42.8 \]
Rounding to one decimal place, we have:
\[ V_{small} \approx 42.9 \text{ cm}^3 \]
So, the volume of the smaller object is 42.9 cm³.
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