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The hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is Rs 5 per 100 cm2.

Answer
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Hint- Here, we will proceed by using the formula for the circumference of any circle of radius r which is given by 2πr and from here we will find the value of radius of the base of the hemispherical dome and then we will use the formula i.e., curved surface area of a hemisphere with base radius r = 2πr2.

Complete step-by-step answer:


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Given, Circumference of the base of dome = 17.6 m
Cost of painting 100 cm2 = Rs 5
As we know that 1 m2 = 10000 cm2
1 cm2=110000 m2100 cm2=(100)(110000) m2100 cm2=1100 m2
So, Cost of painting 1100 m2 = Rs 5
Cost of painting 1 m2 = Rs (5×100) = Rs 500
Let r be the radius of the base of the hemispherical dome as shown in the figure
Also, circumference of any circular surface having radius r = 2πr where π=227
Circumference of the base of dome = 2πr
17.6=2πrr=17.62π=17.62(227)=17.6×744r=2.8 m
So, the radius of the base of the hemispherical dome is 2.8 m
It is given that only the hemispherical dome in the building coloured red in the figure needs to be painted which means only the curved surface area of the hemispherical dome will be painted.
Since, Curved surface area of a hemisphere with base radius r = 2πr2
Curved surface area of the hemispherical dome painted (red colour in figure) = 2(227)(2.8)2
Curved surface area of the hemispherical dome painted (red colour in figure) = 49.28 m2
Also, Total cost of painting the hemispherical dome = (Cost of painting 1 m2)×( Curved surface area of the hemispherical dome painted in m2)
Total cost of painting the hemispherical dome = 500×49.28=Rs 24640
Therefore, the total cost of painting the hemispherical dome of a building is Rs 24640.

Note- In this particular problem, the base of the hemispherical dome is open and is not painted that’s why only the curved surface of the hemisphere is painted. Also, the base of the hemisphere is always a circular surface. Here, it is important to make sure that the units are identical. Here, we have converted every unit into m2.