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Q.

From a solid cylinder whose height is 8 π‘π‘š and radius 6 π‘π‘š, a conical cavity of same height and same base radius is hollowed out. Find the total surface area of the remaining solid. (Take Ο€ = 3. 14)

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Detailed Solution

We have been given the height (8 π‘π‘š) and radius (6 π‘π‘š) of cylinder, whose conical cavity of same height and same base radius is hollowed out. We need to find the total surface area of the remaining solid.

From the given information, we can say that

π»π‘’π‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘–π‘’π‘  π‘œπ‘“ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿ = β„Žπ‘’π‘–π‘”β„Žπ‘‘ π‘Žπ‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘–π‘’π‘  π‘œπ‘“ π‘π‘œπ‘›π‘’

Slant height (𝑙) of a cone can be expressed as  π‘™ = r2 + h2where π‘Ÿ is the radius of the cone and β„Ž is the its height.

l=62+82

𝑙 = 10 cm

Now, π‘†π‘’π‘Ÿπ‘“π‘Žπ‘π‘’ π‘Žπ‘Ÿπ‘’π‘Ž = π‘†π‘’π‘Ÿπ‘“π‘Žπ‘π‘’ π‘Žπ‘Ÿπ‘’π‘Ž π‘œπ‘“ (π‘π‘’π‘Ÿπ‘£π‘’π‘‘ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿ π‘ π‘’π‘Ÿπ‘“π‘Žπ‘π‘’ + π‘π‘œπ‘›π‘’ + π‘‘π‘œπ‘ π‘œπ‘“ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿ)

 We know that, Surface area of cylinder = 2Ο€π‘Ÿβ„Ž 

Surface area of cone = Ο€π‘Ÿπ‘™

Area of top of cylinder= Ο€π‘Ÿ 2

 = 2Ο€π‘Ÿβ„Ž + Ο€π‘Ÿπ‘™ + Ο€π‘Ÿ 2 

= Ο€π‘Ÿ(2β„Ž + 𝑙 + π‘Ÿ)

 = 3. 14 Γ— 6 Γ— (2 Γ— 8 + 10 + 6)

= 602. 88 π‘ π‘ž π‘π‘š

Hence, the total surface area of the remaining solid is 602. 88 π‘ π‘ž π‘π‘š

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From a solid cylinder whose height is 8 and radius 6 , a conical cavity of same height and same base radius is hollowed out. Find the total surface area of the remaining solid. (Take Ο€ = 3. 14)