Q. 14

# The outer and inner diameter of a hollow hemi–spherical vessel are 43 cm and 42 cm. Find the cost of colouring the vessel at 7 paise per square cm.

Answer :

The hemispherical bowl will look like as shown

To find the cost required to paint we first need to calculate its total surface area

Total surface area will include the area of outter curved surface the inner curved surface and the flat surface as shown

Diameter of outer surface = 43 cm

Radius of outer surface = r_{o} = cm = 21.5 cm

Diameter of inner surface = 42 cm

Radius of inner surface = r_{i} = cm = 21 cm

Curved surface area of hemisphere = 2πr^{2}

⇒ outter curved surface area = 2πr_{o}^{2}

= 2 × π ×

= π cm^{2}

= 924.5π cm^{2}

⇒ inner curved surface area = 2πr_{i}^{2}

= 2 × π × 21^{2}

= 882π cm^{2}

The flat area can be found by subtracting the inner circle area from outter circle area

⇒ flat area = πr_{o}^{2} – πr_{i}^{2}

= π(21.5)^{2} – π(21)^{2}

= 462.25π – 441π

= 21.25π cm^{2}

Total surface area of hemispherical vessel = outter curved surface area + inner curved surface area + flat area

⇒ Total surface area of hemispherical vessel = 924.5π + 882π + 21.25π = 1827.75π cm^{2}

Now cost of colouring 1 cm^{2} is 7 paise

So, cost of colouring 1827.75π cm^{2} = 7 × 1827.75 × π

= 7 × 1827.75 ×

= 1827.75 × 22

= 40210.5 paise

100 paise is 1 Rs

⇒ 40210.5 paise = = 402.105 Rs

Therefore, cost of colouring the vessel is 402.105 Rs

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