Math Problem Statement

The two triangular prisms below are similar and have cross-sections that are isosceles triangles. Each edge length in prism Y is 3 times as long as the equivalent edge length in prism X. a) Work out the height of prism Y. b) Calculate the surface area of prism Y. c) How does the surface area of prism Y compare with the surface area of prism X?

Solution

Let's solve each part step by step.

Part (a): Work out the height of prism Y

Since prisms X and Y are similar, the ratio of their corresponding dimensions is the same. The problem states that each edge length in prism Y is 3 times the length of the corresponding edge in prism X. Let's compare the base dimensions of both prisms:

  • Prism X has a length of 12 cm, and prism Y has a length of 36 cm. This confirms the ratio between corresponding edges is 3612=3\frac{36}{12} = 3.

The height of prism X is 10 cm. Therefore, the height of prism Y can be found by multiplying the height of prism X by 3:

Height of prism Y=3×10=30 cm\text{Height of prism Y} = 3 \times 10 = 30 \text{ cm}

So, the height of prism Y is 30 cm.

Part (b): Calculate the surface area of prism Y

Since the prisms are similar, their surface areas are related by the square of the ratio of their corresponding linear dimensions. The surface area of prism Y will be:

Surface area of prism Y=(Length of prism YLength of prism X)2×Surface area of prism X\text{Surface area of prism Y} = \left(\frac{\text{Length of prism Y}}{\text{Length of prism X}}\right)^2 \times \text{Surface area of prism X}

The ratio of corresponding lengths is 3, and the surface area of prism X is given as 800 cm²:

Surface area of prism Y=32×800=9×800=7200 cm2\text{Surface area of prism Y} = 3^2 \times 800 = 9 \times 800 = 7200 \text{ cm}^2

So, the surface area of prism Y is 7200 cm².

Part (c): How does the surface area of prism Y compare with that of prism X?

Since the prisms are similar, the surface areas of similar figures are proportional to the square of the ratio of their corresponding linear dimensions. Here, the ratio of corresponding linear dimensions is 3, so the ratio of their surface areas is 32=93^2 = 9. Therefore, the surface area of prism Y is 9 times that of prism X.

Explanation: The surface area of prism Y is 9 times greater than the surface area of prism X because the ratio of corresponding edge lengths is 3, and surface areas scale by the square of this ratio.


Would you like more details on any part of this problem or have any further questions?

Here are 5 related questions:

  1. What is the volume ratio between two similar prisms?
  2. How does the volume of prism Y compare to that of prism X?
  3. Can the ratio between surface areas of two prisms be less than the ratio between their linear dimensions?
  4. What is the total edge length ratio for similar prisms?
  5. How would the surface area of prism Y change if its edge length ratio to prism X was 4 instead of 3?

Tip: In similar geometric figures, remember that areas scale with the square of the ratio of linear dimensions, while volumes scale with the cube of the ratio.

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Math Problem Analysis

Mathematical Concepts

Similar Figures
Surface Area
Ratio and Proportion

Formulas

Surface Area of Similar Figures: Surface Area of Larger Figure = (Ratio of Corresponding Lengths)^2 * Surface Area of Smaller Figure

Theorems

Similarity Theorem

Suitable Grade Level

Grades 8-10