A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of the solid so formed, - Mathematics

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A cubical block of side 10 cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs. 5 per 100 sq. cm. [Use π = 3.14]

 

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Solution

Side of the cubical block, a = 10 cm
Longest diagonal of the cubical block = a√3 = 10√3 cm
Since the cube is surmounted by a hemisphere, therefore the side of the cube should be equal to the diameter of the hemisphere.
Diameter of the sphere = 10 cm
Radius of the sphere, r = 5 cm
Total surface area of the solid = Total surface area of the cube – Inner cross-section area of the hemisphere + Curved surface area of the hemisphere

`=6a^2-pir^2+2pir^2`

`=6a^2+pir^2`

`=6xx(10)^2+3.14xx5^2`

`=600+78.5=678.5 cm^2`

Total surface area of the solid = 678.5 cm2
Cost of painting 100 cm2 = Rs. 5
Cost of painting 1 cm2 = Rs.5/100
Cost of painting the total surface area of the solid =(5/100)× 678.5 = Rs. 33.925 ≈ Rs. 34.

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2014-2015 (March) All India Set 3

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