Math Problem Statement
Solution
We are given two similar cylinders, and we need to find the surface area of the smaller cylinder. The provided information includes:
- The height of the smaller cylinder = 3 cm.
- The height of the larger cylinder = 5 cm.
- The surface area of the larger cylinder = 236 cm².
Since the cylinders are similar, their surface areas are proportional to the square of the ratio of their heights. The ratio of their heights is:
Now, the ratio of the surface areas will be the square of this ratio:
Using this ratio, we can now find the surface area of the smaller cylinder by multiplying the surface area of the larger cylinder by the ratio:
Let me calculate that.The surface area of the smaller cylinder is approximately when rounded to the nearest tenth.
Do you want further details or have any questions?
Here are 5 related questions for further exploration:
- How do we derive the surface area formula for a cylinder?
- If the heights were different, how would the solution change?
- How can we apply the concept of similarity to other 3D shapes?
- What happens to the surface area if both the height and radius change proportionally?
- How do we find the volume of similar cylinders based on their height ratios?
Tip: When working with similar shapes, remember that surface area scales with the square of the ratio of corresponding lengths, while volume scales with the cube.
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Math Problem Analysis
Mathematical Concepts
Geometry
Surface Area
Similarity of Shapes
Proportionality
Formulas
Surface Area of Cylinder = 2πrh + 2πr²
Ratio of Surface Areas = (Height ratio)²
Theorems
Proportionality Theorem for Similar Shapes
Suitable Grade Level
Grades 8-10
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