Math Problem Statement

What is the volume of a shape composed of a cuboid and a pyramid with given dimensions? The cuboid has a base of 20 cm by 20 cm and a height of 32 cm. The pyramid has a square base of 20 cm by 20 cm and a height of 15 cm. Give the answer in cm³ to 1 decimal place.

Solution

To calculate the volume of the shape, which is made up of a cuboid and a pyramid, we break the calculation into two parts:

Step 1: Volume of the Cuboid

The dimensions of the cuboid are:

  • Length = 20 cm
  • Width = 20 cm
  • Height = 32 cm

The volume of a cuboid is given by: Volume of Cuboid=Length×Width×Height\text{Volume of Cuboid} = \text{Length} \times \text{Width} \times \text{Height} Volume of Cuboid=20×20×32=12,800cm3\text{Volume of Cuboid} = 20 \times 20 \times 32 = 12,800 \, \text{cm}^3


Step 2: Volume of the Pyramid

The base of the pyramid is a square with side 20 cm, and its height is 15 cm.

The volume of a pyramid is given by: Volume of Pyramid=13×Base Area×Height\text{Volume of Pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Height} Base Area=Length×Width=20×20=400cm2\text{Base Area} = \text{Length} \times \text{Width} = 20 \times 20 = 400 \, \text{cm}^2 Volume of Pyramid=13×400×15=13×6,000=2,000cm3\text{Volume of Pyramid} = \frac{1}{3} \times 400 \times 15 = \frac{1}{3} \times 6,000 = 2,000 \, \text{cm}^3


Step 3: Total Volume of the Shape

Total Volume=Volume of Cuboid+Volume of Pyramid\text{Total Volume} = \text{Volume of Cuboid} + \text{Volume of Pyramid} Total Volume=12,800+2,000=14,800cm3\text{Total Volume} = 12,800 + 2,000 = 14,800 \, \text{cm}^3


Final Answer:

The volume of the shape is: 14,800cm3\boxed{14,800 \, \text{cm}^3}

Let me know if you'd like further clarification or additional details.

Questions for exploration:

  1. How would the volume change if the height of the pyramid increased?
  2. What is the surface area of the combined shape?
  3. How do you calculate the volume for shapes with irregular bases?
  4. What is the ratio of the volume of the pyramid to the volume of the cuboid?
  5. How would the calculations differ if the pyramid were a cone?

Tip:

When solving problems involving combined 3D shapes, always separate the calculations for each component and ensure consistency in units!

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Math Problem Analysis

Mathematical Concepts

Volume calculation
3D geometry
Addition of volumes

Formulas

Volume of a cuboid = length × width × height
Volume of a pyramid = (1/3) × base area × height

Theorems

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Suitable Grade Level

Grades 6-8