Math Problem Statement

A and B are two solid prisms that are mathematically similar. The surface area of A is 72.5 cm². The surface area of B is 104.4 cm². The volume of A is 162.5 cm³. Find the volume of B.

Solution

This problem involves two mathematically similar solid prisms, AA and BB. 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 AA is 72.5cm272.5 \, \text{cm}^2
  • The surface area of BB is 104.4cm2104.4 \, \text{cm}^2

The surface areas of similar solids are proportional to the square of their scale factor. Let the scale factor be kk. The ratio of the surface areas is:

Surface area of BSurface area of A=k2\frac{\text{Surface area of } B}{\text{Surface area of } A} = k^2

Substituting the known values:

104.472.5=k2\frac{104.4}{72.5} = k^2

Solving for kk:

k2=104.472.51.44k^2 = \frac{104.4}{72.5} \approx 1.44 k=1.44=1.2k = \sqrt{1.44} = 1.2

Thus, the scale factor kk is 1.21.2.

Step 2: Using the scale factor to find the volume of BB

The volumes of similar solids are proportional to the cube of the scale factor. The volume of AA is given as 162.5cm3162.5 \, \text{cm}^3. So, the volume of BB, denoted as VBV_B, can be found using the formula:

Volume of BVolume of A=k3\frac{\text{Volume of } B}{\text{Volume of } A} = k^3

Substituting the known values:

VB162.5=(1.2)3\frac{V_B}{162.5} = (1.2)^3 VB=162.5×1.728V_B = 162.5 \times 1.728 VB280.32cm3V_B \approx 280.32 \, \text{cm}^3

Final Answer:

The volume of BB is approximately 280.32cm3280.32 \, \text{cm}^3.


Do you need more details or have any questions?

Related questions:

  1. What is the relationship between the areas of two similar shapes?
  2. How would the volume change if the scale factor were doubled?
  3. Can the same principle be applied to non-prismatic shapes?
  4. How can you calculate the height of the prisms given their volumes?
  5. 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