Question

8. From a solid cylinder whose height is 2.4 crn and diameter 1.4 cm, a conical cavity of the
same height and same diameter is hollowed out. Find the total surface area of the
remaining solid to the nearest cm.
​

Answer 1

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

• From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm²

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

• Height of the conical part = Height of the cylindrical part = 2.4cm

• Diameter of the cylindrical part = 1.4 cm

$\underline{\underline{\sf{\maltese\:\:To\:Find}}}$

• Total surface area of the remaining solid to the nearest cm²

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

• Total surface area of remaining solid = 17.6 cm²

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

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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• Since cylinder is solid, it has a base also whose area is also to be calculated.

Total surface area of remaining solid = Curved Surface Area of cylinder + Area of cylinder base + Curved surface area of cone

⇒ Total surface area of remaining solid

= 2πrh + πr² + πrl

Since r = diameter/2

= [2 × π × (diameter/2) × h] + [π × (diameter/2)²] + [π × (diameter/2) × l]

= [2 × π × (1.4 cm/2) × h] + [π × (1.4 cm/2)²] + [π × (1.4 cm/2) × l]

= [2 × π × 0.7 cm × h] + [π × (0.7 cm)²] + [π × 0.7 cm × l]

Since π = 22/7

= [2 × 22/7 × 0.7 cm × h] + [22/7 × 0.49 cm²] + [22/7 × 0.7 cm × l]

= [2 × 22/1 × 0.1 cm × h] + [22/1 × 0.07 cm²] + [22/1 × 0.1 cm × l]

= [44 × 0.1 cm × h] + [1.54 cm²] + [2.2 cm × l]

= [4.4 cm × h] + [1.54 cm²] + [2.2 cm × l]

Given Height = 2.4 cm

= [4.4 cm × 2.4 cm] + [1.54 cm²] + [2.2 cm × l]

= 10.56 cm² + 1.54 cm² + [2.2 cm × l]

= 12.1 cm² + [2.2 cm × l]

Since l² = h² + r² ⇒ l = √(h² + r²) . Note "l" is Slant Height

= 12.1 cm² + [2.2 cm × √(h² + r²)]              

= 12.1 cm² + [2.2 cm × √{(2.4 cm)² + (0.7 cm)²}]    

= 12.1 cm² + [2.2 cm × √{5.76 cm² + 0.49 cm²}]    

= 12.1 cm² + [2.2 cm × √{6.25 cm²}]    

= 12.1 cm² + [2.2 cm × {2.5 cm}]    

= 12.1 cm² + [5.5 cm²]    

= 17.6 cm²    

  

⇒ Total surface area of remaining solid = 17.6 cm²    

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