Question

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.​

Answer 1

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• A hemispherical depression is cut out from one face of the cubical wooden block such that the diameter (I) of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid

$\underline{\underline{\sf{\maltese\:\:Given}}}$

• Diameter of Hemisphere = l

• Diameter (I) of the hemisphere is equal to the edge of the cube.

• A hemisphere of radius 1/2 is cut out from the top face of the cuboidal wooden block

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• The surface area of the remaining solid

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• Surface Area of Solid =  l²/4 + (24 + π)/4

$\underline{\underline{\sf{\maltese\:\:Calculations}}}$

First we need to know about some basic terms before going into answer

• Diameter : The diameter is the length of the line through the center that touches two points on the edge of the circle.

• Radius : The distance from the center of the circle to any point on the circle

Also :

• Diameter = 2 × Radius
• Radius = Diameter/2

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As already given :

• Diameter (I) of the hemisphere is equal to the edge of the cube

Diameter of the hemispher = Side of Cube

From This :

• Diameter of the hemispher = l

• Side of Cube = l

Here, the base of the hemisphere would not be included in the total surface area of the wooden cube.

Surface Area of Solid = Area of Cube + Curved Surface Area of hemisphere - Base Area of hemisphere

⇒ Surface Area of Solid = 6 × (Side)² + Curved Surface Area of hemisphere - Base Area of hemisphere

⇒ Surface Area of Solid = 6 × (l)² + Curved Surface Area of hemisphere - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + Curved Surface Area of hemisphere - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + 2πr² - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + 2 × π × (diameter/2)² - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + 2 × π × (l/2)² - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + 2 × π × l²/4 - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + 2 × π × l²/(2 × 2) - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + π × l²/2 - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + (πl²)/2 - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + (πl²)/2 - Base Area of hemisphere

⇒ Surface Area of Solid = 6l² + (πl²)/2 - πr²

⇒ Surface Area of Solid = 6l² + (πl²)/2 - π × (diameter/2)²

⇒ Surface Area of Solid = 6l² + (πl²)/2 - π × (l/2)²

⇒ Surface Area of Solid = 6l² + (πl²)/2 - π × l²/4

⇒ Surface Area of Solid = 6l² + (πl²)/2 - (πl²)/4

⇒ Surface Area of Solid = 6l² + (πl² × 2)/2 × 2 - (πl²)/4

⇒ Surface Area of Solid = 6l² + (πl² × 2)/4 - (πl²)/4

⇒ Surface Area of Solid = 6l² + [(πl² × 2)- (πl²)]/4

⇒ Surface Area of Solid = 6l² + (2πl² - πl²)/4

⇒ Surface Area of Solid = 6l² + (πl²)/4

⇒ Surface Area of Solid = l² + (6 + π/4)

⇒ Surface Area of Solid = l² + (6 × 4+ π)/4

⇒ Surface Area of Solid = l² + (24 + π)/4

⇒ Surface Area of Solid = 1/4 × l² + (24 + π)/4

⇒ Surface Area of Solid =  l²/4 + (24 + π)/4

Note We got :

• Area of Cube = 6l²

• Curved Surface Area of Hemisphere = (πl²)/2

• Base Area of Hemisphere = (πl²)/4

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Answer 2

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Now, the diameter of hemisphere = Edge of the cube = l

So, the radius of hemisphere = l/2

∴ The total surface area of solid = surface area of cube + CSA of hemisphere – Area of base of hemisphere

TSA of remaining solid = 6 (edge)2+2πr2-πr2

= 6l2 πr2

= 6l2+π(l/2)2

= 6l2+πl2/4

= l2/4(24+π) sq. units