Question :The boilers are used in thermal power plants to store water and then used to produce steam. One such boiler consists of a cylindrical part in middle and two hemispherical parts at its both ends.
Length of the cylindrical part is 7m and radius of
cylindrical part is m
Find the total surface area and the volume of the boiler. Also, find the ratio of the volume of cylindrical part to the volume of one hemispherical part.
The correct answer :The boiler consists of a cylindrical part and two hemispherical parts. Let’s first find the dimensions of each part:
Cylindrical part:
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Length = 7 m
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Radius = 1 m
Hemispherical part:
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Radius = 1 m
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Total surface area of the boiler: The total surface area of the boiler can be found by adding the surface areas of the cylindrical and hemispherical parts. The formula for the surface area of a cylinder is 2πrh + 2πr^2, where r is the radius and h is the height, and the formula for the surface area of a hemisphere is 2πr^2.
Surface area of cylindrical part = 2π(1)(7) + 2π(1)^2 = 14π + 2π = 16π Surface area of one hemispherical part = 2π(1)^2 = 2π
Total surface area of the boiler = Surface area of cylindrical part + Surface area of two hemispherical parts = 16π + 2(2π) = 20π
Therefore, the total surface area of the boiler is 20π square meters.
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Volume of the boiler: The volume of the boiler can be found by adding the volumes of the cylindrical and hemispherical parts. The formula for the volume of a cylinder is πr^2h, where r is the radius and h is the height, and the formula for the volume of a hemisphere is 2/3πr^3.
Volume of cylindrical part = π(1)^2(7) = 7π Volume of one hemispherical part = 2/3π(1)^3 = 2/3π
Volume of the boiler = Volume of cylindrical part + Volume of two hemispherical parts = 7π + 2(2/3π) = 7π + 4/3π = 25/3π
Therefore, the volume of the boiler is 25/3π cubic meters.
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Ratio of the volume of cylindrical part to the volume of one hemispherical part: The ratio of the volume of the cylindrical part to the volume of one hemispherical part can be found by dividing the volume of the cylindrical part by the volume of one hemispherical part.
