Find the ratio of the volumes of a cone, hemisphere, and cylinder which are made with the same circular base and height.

Points to Remember:

• The volumes of a cone, hemisphere, and cylinder with the same circular base and height are related through simple formulas.
• The ratio will be a constant regardless of the specific dimensions of the base and height.
• Understanding the formulas for the volume of each 3D shape is crucial.

Introduction:

This question requires a factual and analytical approach to determine the ratio of the volumes of a cone, hemisphere, and cylinder sharing the same circular base and height. The volume of each shape is calculated using specific formulas based on its dimensions. Let’s assume the radius of the common circular base is ‘r’ and the height of the cylinder and cone is ‘h’. Since the hemisphere is half a sphere, its height will also be ‘r’.

Body:

1. Volume Formulas:

• Cone: The volume of a cone is given by the formula: V_cone = (1/3)πr²h
• Hemisphere: The volume of a hemisphere is given by the formula: V_hemisphere = (2/3)πr³ (Note: h = r for a hemisphere)
• Cylinder: The volume of a cylinder is given by the formula: V_cylinder = πr²h

2. Calculating the Ratio:

Since the cone and cylinder share the same base and height (r and h), we can directly compare their volumes. However, for the hemisphere, the height is equal to the radius (h = r). Therefore, we will use ‘r’ instead of ‘h’ in the hemisphere’s volume formula for consistency in the ratio calculation.

Let’s find the ratio: V_cone : V_hemisphere : V_cylinder

Substituting the formulas:

(1/3)πr²h : (2/3)πr³ : πr²h

Since h=r for the hemisphere, we can rewrite the ratio as:

(1/3)πr³ : (2/3)πr³ : πr³

Now, we can simplify the ratio by dividing each term by πr³:

(1/3) : (2/3) : 1

Multiplying each term by 3 to remove fractions:

1 : 2 : 3

3. Interpretation of the Ratio:

The ratio of the volumes of the cone, hemisphere, and cylinder with the same circular base and height (where the height of the hemisphere is equal to its radius) is 1:2:3. This means that the volume of the cylinder is three times the volume of the cone and 1.5 times the volume of the hemisphere.

Conclusion:

In conclusion, the ratio of the volumes of a cone, hemisphere, and cylinder with the same circular base and height is 1:2:3. This ratio remains constant irrespective of the specific values of the radius and height. This understanding is crucial in various applications, from engineering design to architectural calculations. This analysis highlights the fundamental geometric relationships between these three-dimensional shapes. Further exploration could involve examining the surface area ratios of these shapes or exploring similar relationships with other geometric solids. This knowledge promotes a deeper understanding of spatial reasoning and mathematical principles.

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