https://file.aiquickdraw.com/mathbot/file/1727110698867ng1t4e6x.jpg

The problem involves two mathematically similar ornaments, with the task of finding the surface area of the smaller ornament, given the surface area of the larger one.

### Approach:
For geometrically similar objects, the ratio of their surface areas is proportional to the square of the ratio of their corresponding linear dimensions.

Given:
- Surface area of the larger ornament: \( 21,840 \, \text{cm}^2 \)
- Height of the larger ornament: \( 140 \, \text{cm} \)
- Height of the smaller ornament: \( 35 \, \text{cm} \)

The formula to calculate the surface area of the smaller ornament based on the similarity ratio is:

\[
\frac{\text{Surface area of smaller ornament}}{\text{Surface area of larger ornament}} = \left( \frac{\text{Height of smaller ornament}}{\text{Height of larger ornament}} \right)^2
\]

Substituting the given values:

\[
\frac{\text{Surface area of smaller ornament}}{21,840} = \left( \frac{35}{140} \right)^2
\]

Let me calculate this for you.The surface area of the smaller ornament is \( 1365.0 \, \text{cm}^2 \).

Would you like further clarification or additional details on the process?

Here are five related questions to expand on this:

1. What is the relationship between volume and height for similar geometric objects?
2. How would the result change if the height of the smaller ornament were halved again?
3. What would be the formula if we needed to calculate the volume instead of the surface area?
4. Can we apply the same concept to non-geometrically similar shapes?
5. What other real-world examples use the concept of mathematical similarity?

**Tip**: For mathematically similar objects, the surface area scales with the square of the linear dimension, while the volume scales with the cube.

The ornaments below are mathematically similar. Calculate the surface area of the smaller ornament. If your answer is a decimal, give it to 1 d.p.

Learn how to calculate the surface area of a smaller ornament based on mathematical similarity. This problem involves using the ratio of heights to find the surface area of a mathematically similar object. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation