Question

A cubical block of side 7cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.​

Answer 1

$\huge\tt{★\:Solution\: :-}$

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$\tt{The\: greatest \:diameter\:=\:side }$$\tt{of\: the\: cube\:=\:7cm.}$

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$\tt{The\: radius \:of \:the\: hemisphere \:=\:3.5cm.}$

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$\tt{Now,}$

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$\tt{The\: Surface\: Area \:of \:Solid=}$$\tt{Surface \:Area \:of \:Cube\: - Surface\: Area \:of }$$\tt{Base\: Hemisphere\: +\: Curve }$$\tt{Surface\: Area \:of \:Hemisphere.}$

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$\tt{Surface\: Area \:of\: Cube}$

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↦$\tt{6\:༝}$$\tt{side}^{2}$

↦$\tt{6\:×\:7\:×\:7}$

↦$\tt{294\:sq \:cm}$

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$\tt{Surface\: Area \:of \:Base\: of \:Hemisphere\: is,}$

↦$\tt{π}$$\tt{r}^{2}$

↦$\tt\frac{22}{7}$$\tt{༝}{3.5}^{2}$

↦$\tt{38.5\:cm}^{2}$

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$\tt{Curved\: Surface \:Area \:of\: Hemisphere \:is,}$

↦$\tt{2\:×\:38.5}$

↦$\tt{77\:sq \:cm}$

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$\tt{∴ \:total\: Surface\: Area \:is\: :- }$

↦$\tt{294\:−\:38.5}$$\tt{+\:77}$

↦$\tt{332.5\:sq \:cm.}$

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$\tt\blue{Answer\:↠}$$\tt\green{332.5\:sq \:cm.}$

Answer 2

Given :

• Each side of cube = 7 cm
• Diameter of hemisphere = 7 cm

To Find :

• Total Surface are of solid = ?

Solution :

⚽ First of all we will find the area of the cube :

➻ Area of cube = 6 × (side)²

➻ Area of cube = 6 × (7)²

➻ Area of cube = 6 × 49

➻ Area of cube = 294 cm²

• Hence,the area of the cube is 294 cm².

⚽ Now,let's find the radius of the hemisphere :

➺ Radius of hemisphere = Diameter of hemisphere ÷ 2

➺ Radius of hemisphere = 7 ÷ 2

➺ Radius of hemisphere = 7/2 cm

• Hence,the radius of the hemisphere is 7/2 cm.

⚽ Finding curved surface area of the hemisphere :

→ CSA of hemisphere = 2πr²

→ CSA of hemisphere = 2 × 22/7 × 7/2 × 7/2

→ CSA of hemisphere = 11 × 7

→ CSA of hemisphere = 77 cm²

• Hence,the curved surface area of the hemisphere is 77 cm².

⚽ Finding Base area of the hemisphere :

⋙ Base area of hemisphere = πr²

⋙ Base area of hemisphere = 22/7 × 7/2 × 7/2

⋙ Base area of hemisphere = 11 × 7/2

⋙ Base area of hemisphere = 11 × 3.5

⋙ Base area of hemisphere = 38.5 cm²

• Hence,the base area of the hemisphere is 38.5 cm².

⚽ Now, let's calculate the total surface area of the cone :

⊶ TSA of solid = Area of cube + CSA of hemisphere - Base area of hemisphere

⊶ TSA of solid = 294 + 77 - 38.5

⊶ TSA of solid = 371 - 38.5

⊶ TSA of solid = 332.5 cm²

• Hence,the total surface area of the the solid is 332.5 cm².