The two pasta jars are mathematically similar, meaning their dimensions are proportional. To find the volume of the smaller jar, we can use the concept of similarity ratios.

Step 1: Set up the similarity ratio

The height of the large jar is 30 cm, and the height of the small jar is 24 cm. The ratio of their linear dimensions is:

$\text{Linear Ratio} = \frac{\text{Height of Small Jar}}{\text{Height of Large Jar}} = \frac{24}{30} = \frac{4}{5}.$

Step 2: Use the volume ratio

The volumes of similar shapes are proportional to the cube of their linear dimensions. Therefore, the volume ratio is:

$\text{Volume Ratio} = \left(\frac{\text{Linear Ratio}}{\text{Height}}\right)^3 = \left(\frac{4}{5}\right)^3 = \frac{64}{125}.$

Step 3: Find the volume of the smaller jar

The volume of the large jar is $1.6 \, \text{L} = 1600 \, \text{cm}^3$. Using the volume ratio:

$\text{Volume of Small Jar} = \text{Volume of Large Jar} \times \text{Volume Ratio}$

$\text{Volume of Small Jar} = 1600 \times \frac{64}{125} = 819.2 \, \text{cm}^3 = 0.8192 \, \text{L}.$

Final Answer:

The volume of the smaller jar is approximately 0.819 L.

Would you like a step-by-step breakdown of any specific part? 😊

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1. What does "mathematically similar" mean in terms of geometry?
2. How does the volume of a shape change with scale factors?
3. Why is the volume proportional to the cube of the linear dimensions?
4. Could we solve the problem if the jars were not mathematically similar?
5. How would the calculation differ if the jar dimensions were given in meters?

Tip: Always remember, for similar shapes, surface areas are proportional to the square of the linear ratio, and volumes are proportional to the cube.