Math Problem Statement
Solution
This problem involves two mathematically similar solid prisms, and . Since the prisms are similar, their surface areas and volumes will follow specific proportional relationships.
Step 1: Finding the scale factor for surface area
Given that:
- The surface area of is
- The surface area of is
The surface areas of similar solids are proportional to the square of their scale factor. Let the scale factor be . The ratio of the surface areas is:
Substituting the known values:
Solving for :
Thus, the scale factor is .
Step 2: Using the scale factor to find the volume of
The volumes of similar solids are proportional to the cube of the scale factor. The volume of is given as . So, the volume of , denoted as , can be found using the formula:
Substituting the known values:
Final Answer:
The volume of is approximately .
Do you need more details or have any questions?
Related questions:
- What is the relationship between the areas of two similar shapes?
- How would the volume change if the scale factor were doubled?
- Can the same principle be applied to non-prismatic shapes?
- How can you calculate the height of the prisms given their volumes?
- What are other geometric properties that change with the scale factor?
Tip: When dealing with similar solids, always remember that surface areas are proportional to the square of the scale factor, while volumes are proportional to the cube of the scale factor.
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Math Problem Analysis
Mathematical Concepts
Similar Solids
Surface Area and Volume Proportions
Scale Factor
Formulas
Surface Area Ratio: (Surface Area of B) / (Surface Area of A) = k²
Volume Ratio: (Volume of B) / (Volume of A) = k³
Theorems
Proportionality Theorem for Similar Solids
Suitable Grade Level
Grades 8-10
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