Math Problem Statement
Solution
To find the volume of the frustum in the given problem, we can follow this approach:
Formula for the volume of a frustum of a pyramid:
The volume of a frustum of a pyramid is given by: where:
- is the height of the frustum,
- is the area of the base of the larger section,
- is the area of the base of the smaller section.
Step-by-step calculation:
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Dimensions from the diagram:
- The base of the larger pyramid section is a rectangle with sides and , so its area is:
- The base of the smaller pyramid section (top) is a rectangle with sides and , so its area is:
- The height of the frustum is given as .
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Plug these values into the formula: First, calculate : Now, substitute this back into the volume formula: Simplify the expression inside the parentheses:
Final Answer:
The volume of the frustum is .
Would you like more details on any step or have any further questions?
Here are five related questions you might want to explore:
- How would the formula change if the frustum had a triangular base?
- What would the volume be if the smaller pyramid was removed entirely?
- How does the volume of the original pyramid compare to that of the frustum?
- How can the formula be adapted for a frustum with non-parallel sides?
- What real-world applications can this volume calculation method be used for?
Tip: Always double-check units to ensure accuracy when performing calculations like volume!
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Math Problem Analysis
Mathematical Concepts
Geometry
Volume of Frustum
Pyramids
Formulas
Volume of Frustum: V = (1/3) * h * (A1 + A2 + sqrt(A1 * A2))
Area of Rectangle: A = length * width
Theorems
Frustum Volume Theorem
Suitable Grade Level
Grades 8-10
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