Math Problem Statement

Find the volume of the following shapes based on their respective formulas.

Solution

Let's go through the volumes of each figure from the image.

  1. Cube (1st image)
    The volume of a cube is calculated by the formula:
    V=a3V = a^3 where aa is the length of one side of the cube.
    Since there are no dimensions shown in the image for this cube, please provide the side length to calculate the volume.

  2. Rectangular Prism (2nd image)
    The volume of a rectangular prism is calculated by the formula:
    V=l×w×hV = l \times w \times h where ll is the length, ww is the width, and hh is the height.
    We can calculate the volume once the dimensions are provided.

  3. Cube (3rd image)
    This looks like another cube, and the side is given as 12 cm.
    Using the formula for a cube, we get: V=123=12×12×12=1728cm3V = 12^3 = 12 \times 12 \times 12 = 1728 \, \text{cm}^3

  4. Combined cubes (4th image)
    This figure seems to consist of multiple cubes. If you can provide the size of each cube, we can calculate the total volume by summing the volumes of each individual cube.

  5. Sphere (5th image)
    The volume of a sphere is calculated by the formula:
    V=43πr3V = \frac{4}{3} \pi r^3 where rr is the radius.
    Given r=10mmr = 10 \, \text{mm}, we calculate:
    V=43π(10)3=43π×1000=40003π4188.79mm3V = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi \times 1000 = \frac{4000}{3} \pi \approx 4188.79 \, \text{mm}^3

  6. Cylinder (6th image)
    The volume of a cylinder is calculated by the formula:
    V=πr2hV = \pi r^2 h Given d=21dmd = 21 \, \text{dm} and h=50dmh = 50 \, \text{dm}, we first find the radius:
    r=d2=212=10.5dmr = \frac{d}{2} = \frac{21}{2} = 10.5 \, \text{dm} Then, calculate the volume:
    V=π(10.5)2×50=π×110.25×5017362.5dm3V = \pi (10.5)^2 \times 50 = \pi \times 110.25 \times 50 \approx 17362.5 \, \text{dm}^3

  7. Cone (7th image)
    The volume of a cone is calculated by the formula:
    V=13πr2hV = \frac{1}{3} \pi r^2 h Given r=2.8cmr = 2.8 \, \text{cm} and h=6.5cmh = 6.5 \, \text{cm}, we calculate:
    V=13π(2.8)2×6.5=13π×7.84×6.553.98cm3V = \frac{1}{3} \pi (2.8)^2 \times 6.5 = \frac{1}{3} \pi \times 7.84 \times 6.5 \approx 53.98 \, \text{cm}^3

Let me know if you'd like to go into more detail or if you have any other questions! Here's a quick tip for understanding volume:

Tip: Always ensure the units are consistent when calculating volumes—convert them if needed before using the formulas.

Would you like to dive deeper into any of these, or do you have other questions related to volume calculations?

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Math Problem Analysis

Mathematical Concepts

Volume of 3D shapes
Cube
Rectangular Prism
Sphere
Cylinder
Cone

Formulas

Volume of Cube: V = a^3
Volume of Rectangular Prism: V = l * w * h
Volume of Sphere: V = (4/3) * π * r^3
Volume of Cylinder: V = π * r^2 * h
Volume of Cone: V = (1/3) * π * r^2 * h

Theorems

Volume calculations of 3D objects

Suitable Grade Level

Grades 7-9