https://r2.erweima.ai/mathbot/image/1719245270929ynn9n3nf.png

Given that the glass objects are mathematically similar, we can use the properties of similar objects to find the height \( h \) of object C.

### Step-by-Step Solution:

1. **Ratios of Similar Objects**:
   - For similar objects, the ratio of corresponding lengths (linear dimensions) is constant.
   - The ratio of surface areas is the square of the ratio of corresponding lengths.
   - The ratio of volumes is the cube of the ratio of corresponding lengths.

2. **Determine the Scale Factor**:
   - We know the volume of object B and object C:
     \[
     \text{Volume of B} = 300 \, \text{cm}^3
     \]
     \[
     \text{Volume of C} = 2400 \, \text{cm}^3
     \]

   - Let the scale factor for the volumes be \( k^3 \):
     \[
     k^3 = \frac{\text{Volume of C}}{\text{Volume of B}} = \frac{2400}{300} = 8
     \]
     \[
     k = \sqrt[3]{8} = 2
     \]

3. **Find the Height of Object C**:
   - Since the height of object B is not given directly, we first find the height ratio using the height of object A (which is given).
   - The height of object A is 3 cm. We need the height of object B first, since B and C are similar and we have the volumes and surface areas for B.
   - The volume ratio for A to B can be used:
     \[
     k_{AB}^3 = \frac{\text{Volume of B}}{\text{Volume of A}}
     \]
     Assuming A's volume (since it's not provided, we'll denote it \( V_A \) ) and given that \( k_{AB} \) (linear scale factor) can be calculated from surface area.
     \[
     k_{AB}^2 = \frac{\text{Surface Area of B}}{\text{Surface Area of A}} = \frac{180}{20} = 9
     \]
     \[
     k_{AB} = \sqrt{9} = 3
     \]
     So the linear scale factor from A to B is 3, thus the height of B:
     \[
     \text{Height of B} = 3 \times 3 = 9 \, \text{cm}
     \]

   - Now we use the linear scale factor \( k = 2 \) between B and C to find the height of C:
     \[
     \text{Height of C} = 9 \times 2 = 18 \, \text{cm}
     \]

Therefore, the height \( h \) of object C is \( \boxed{18.00 \, \text{cm}} \).

Would you like more details on any part of this solution? Here are some related questions you can ask:

1. How do you calculate the surface area and volume ratios for similar objects?
2. Can you explain the concept of scale factors in similar figures?
3. How do you find the linear dimensions of similar objects given their surface area and volume?
4. What are the properties of similar geometric shapes?
5. How do you solve problems involving proportions and similar figures?

**Tip:** When dealing with similar objects, always remember that the ratios of corresponding linear dimensions, areas, and volumes are related as the square and cube of the linear dimension ratio, respectively.

Learn how to calculate the height of object C using the properties of similar figures and volume ratios. This problem involves understanding geometric similarity, scale factors, and applying volume and surface area ratios. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation