Question

Q. The interior of a building is in the form of a right circular cylinder of diameter 4.2 m and height 4 m surrounded by a cone. The vertical height of the cone is 2.1 m. Find the outer surface area and volume of the building. ( Use $\pi = \frac{22}{7}$)
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Answer 1

Question :

The interior of a building is in the form of a right circular cylinder of a diameter 4.2 m and height  4 m, surmounted by a cone. The vertical height of the cone is 2.1 m. Find the outer surface area and volume of the building

Given :

• Hieght of Cylinder = 4m
• Diameter of Cylinder = 4.2m
• Hieght of Cone = 2.1m
• Diameter of Cone = 4.2m

To Find :

• Outer surface area of building?
• Volume of the building?

Formula To Be Applied:

We will use the formula of CSA and volume of Cylinder:

• Volume = pi(r²h)
• CSA = 2(pi)rh

We will use the formula of LSA and volume of Cone :

• Volume = pi(r²)h/3
• LSA = (pi)rl

Solution :

Given, the Cylinder is surmounted by a cone .So the volume of building is the sum of volume of cone and Cylinder.

${\implies\displaystyle \rm Volume_{building} = \displaystyle \rm Volume_{cone} + \displaystyle \rm Volume_{cylinder}}$

But, Outer surface area of building is the sum of LSA of Cone and CSA of cylinder

${ \implies \displaystyle \rm osa_{building} = \displaystyle \rm csa_{cylinder} + \displaystyle \rm lsa_{cone}}$

Here, OSA = outer surface area

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1] Volume Of Cylinder :

Given, Diameter of base = 4.2m

Hieght = 4m

We know that :

Radius = d/2

or, r = 2.1 cm

So, Volume = pi(r²h)

${ \implies\displaystyle \rm Volume_{cylinder} = \pi \times 2.1 \times 2.1 \times 4}$

${\implies \displaystyle \rm Volume_{cylinder} = \frac{22}{7} \times 2.1 \times 2.1 \times 4}$

${\implies \displaystyle \rm Volume_{cylinder} = 22 \times 0.3 \times 2.1 \times 4}$

$\red{ \underline{ \boxed{ \therefore\displaystyle \rm Volume_{cylinder} = 55.44 {m}^{3} }}}$

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2] Volume Of Cone :

Given, Diameter of Cone = 4.2m

Hieght = 2.1 cm

We know that :

Radius = d/2

or, r = 2.1cm

So, Volume = (pi)r²h/3

${ \implies \displaystyle \rm Volume_{cone} = \pi \times 2.1 \times 2.1 \times 2.1 \times \frac{1}{3} }$

${ \implies \displaystyle \rm Volume_{cone} = \frac{22}{7} \times 2.1 \times 2.1 \times 2.1 \times \frac{1}{3} }$

${ \implies \displaystyle \rm Volume_{cone} = \frac{22}{7} \times 2.1 \times 2.1 \times 0.7}$

${ \implies\displaystyle \rm Volume_{cone} = 22 \times 2.1 \times 2.1 \times 0.1}$

${ \implies \displaystyle \rm Volume_{cone} = 2.2 \times 2.1 \times 2.1}$

$\blue{ \underline{ \boxed{ \therefore \displaystyle \rm Volume_{cone} = 9.70 {m}^{3} }}}$

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3] CSA of Cylinder :

Radius of cylinder = 2.1m

Hieght = 4m

So, CSA = 2pi(rh)

${\implies \displaystyle \rm csa_{cylinder} = 2 \times \frac{22}{7} \times 2.1 \times 4}$

${ \implies \displaystyle \rm csa_{cylinder} = 2 \times 22 \times 0.3 \times 4}$

$\implies \displaystyle \rm csa_{cylinder} = 44 \times 1.2$

$\green{ \underline{ \boxed{\implies \displaystyle \rm csa_{cylinder} = 52.8 {m}^{2} }}}$

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4] LSA of Cone :

Radius of cone = 2.1m

Hieght = 2.1m

So, Lateral Lenght² = R²+H²

or, L² = 2.1²+2.1²

or, L² = 2(2.1)²

or, L = 2.1√2

so, LSA = (pi)rl

$\implies \displaystyle \rm lsa_{cone} = \frac{22}{7} \times 2.1 \times 2.1 \sqrt{2}$

$\implies \displaystyle \rm lsa_{cone} = 22 \times 0.3 \times 2.1 \sqrt{2}$

$\pink{ \underline{ \boxed{ \therefore \displaystyle \rm lsa_{cone} = 19.60 {m}^{2} }}}$

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5] Volume Of The Building:

As we had already stated that :

${\implies\displaystyle \rm Volume_{building} = \displaystyle \rm Volume_{cone} + \displaystyle \rm Volume_{cylinder}}$

${\implies\displaystyle \rm Volume_{building} = \displaystyle \rm9.70 {m}^{3} + \displaystyle \rm55.44 {m}^{3} }$

$\orange{ \underline{ \boxed{ \therefore {\displaystyle \rm Volume_{building} =65.14 {m}^{3} }}}}$

Hence, Volume of building = 65.14m³

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6] OSA of Building :

as we have already stated that :

${ \implies \displaystyle \rm osa_{building} = \displaystyle \rm csa_{cylinder} + \displaystyle \rm lsa_{cone}}$

${ \implies \displaystyle \rm osa_{building} = \displaystyle \rm 52.8 {m}^{2} + \displaystyle \rm 19.6 {m}^{2} }$

$\purple { \underline{ \boxed{ \therefore \displaystyle \rm osa_{building} = 72.4 {m}^{2} }}}$

Hence, OSA of Building = 72.4 m²