Math Problem Statement

Work out the volume of the frustum of a rectangular-based pyramid with the following dimensions: larger base (7 cm by 14 cm), smaller base (3 cm by 6 cm), and height (12 cm).

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 VV of a frustum of a pyramid is given by: V=13h(A1+A2+A1A2)V = \frac{1}{3} h \left( A_1 + A_2 + \sqrt{A_1 \cdot A_2} \right) where:

  • hh is the height of the frustum,
  • A1A_1 is the area of the base of the larger section,
  • A2A_2 is the area of the base of the smaller section.

Step-by-step calculation:

  1. Dimensions from the diagram:

    • The base of the larger pyramid section is a rectangle with sides 7cm7 \, \text{cm} and 14cm14 \, \text{cm}, so its area A1A_1 is: A1=7×14=98cm2A_1 = 7 \times 14 = 98 \, \text{cm}^2
    • The base of the smaller pyramid section (top) is a rectangle with sides 3cm3 \, \text{cm} and 6cm6 \, \text{cm}, so its area A2A_2 is: A2=3×6=18cm2A_2 = 3 \times 6 = 18 \, \text{cm}^2
    • The height of the frustum is given as 12cm12 \, \text{cm}.
  2. Plug these values into the formula: V=13×12(98+18+98×18)V = \frac{1}{3} \times 12 \left( 98 + 18 + \sqrt{98 \times 18} \right) First, calculate 98×18\sqrt{98 \times 18}: 98×18=1764=42\sqrt{98 \times 18} = \sqrt{1764} = 42 Now, substitute this back into the volume formula: V=13×12(98+18+42)V = \frac{1}{3} \times 12 \left( 98 + 18 + 42 \right) Simplify the expression inside the parentheses: V=13×12×158V = \frac{1}{3} \times 12 \times 158 V=4×158=632cm3V = 4 \times 158 = 632 \, \text{cm}^3

Final Answer:

The volume of the frustum is 632cm3632 \, \text{cm}^3.

Would you like more details on any step or have any further questions?

Here are five related questions you might want to explore:

  1. How would the formula change if the frustum had a triangular base?
  2. What would the volume be if the smaller pyramid was removed entirely?
  3. How does the volume of the original pyramid compare to that of the frustum?
  4. How can the formula be adapted for a frustum with non-parallel sides?
  5. 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