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A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere of same radius. If the radius of the hemisphere is $4.2$ cm and the total height of the toy is $10.2$ cm, find the volume of the wooden toy. Also, find the total surface area of the toy.
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Height of conical part $= 10.2 - 4.2 = 6$ cm
Volume of toy = Volume of conical part + Volume of hemispherical part
$= (\frac{1}{3} \times \frac{22}{7} \times (4.2)^2 \times 6) + (\frac{2}{3} \times \frac{22}{7} \times (4.2)^3)$
$= 266.112$
Hence, Volume of toy is $266.112$ cm$^3$
Slant height of conical part $= \sqrt{(4.2)^2 + (6)^2} \approx 7.32$ cm
TSA of the toy = CSA of hemispherical part + CSA of conical part
$= (2 \times \frac{22}{7} \times (4.2)^2) + (\frac{22}{7} \times 4.2 \times 7.32)$
$= 207.504$
Hence, TSA of toy is $207.504$ cm$^2$
Volume of toy = Volume of conical part + Volume of hemispherical part
$= (\frac{1}{3} \times \frac{22}{7} \times (4.2)^2 \times 6) + (\frac{2}{3} \times \frac{22}{7} \times (4.2)^3)$
$= 266.112$
Hence, Volume of toy is $266.112$ cm$^3$
Slant height of conical part $= \sqrt{(4.2)^2 + (6)^2} \approx 7.32$ cm
TSA of the toy = CSA of hemispherical part + CSA of conical part
$= (2 \times \frac{22}{7} \times (4.2)^2) + (\frac{22}{7} \times 4.2 \times 7.32)$
$= 207.504$
Hence, TSA of toy is $207.504$ cm$^2$