Question

A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome

is 22.8m, find the cost of painting it, given the cost of painting is Rs. 7 per 100 cm​

Answer 1

Answer :-

$\: \\ \large{\underline{\underline{\sf{\leadsto \: \;\; Firstly, \; let's\; understand\; the \; concept \; used\; :-}}}}$

Here the concept of Circumference of Circle and CSA of Hemisphere is used. We know that if we are given the circumference of circular plane, we can find our its radius. Also, after finding the radius we can apply it into the formula of CSA of Hemisphere because dome is hemisphere in shape and for painting, we would neglect the base. Also we need to change the unit 7 per 100 cm to m².

Let's do it !!

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★ Formula Used :-

$\: \\ \large{\boxed{\boxed{\sf{Circumference\; of \; Circle_{(base\; of \; Dome)} \: \; = \; \bf{2 \pi r}}}}}$

$\: \\\large{\boxed{\boxed{\sf{CSA \; of \; Hemisphere_{(Dome)} \:\; = \:\; \bf{2 \pi r^{2}}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{1 \; m \: = \:\; \bf{100\; cm}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{1 \: m^{2} \; = \;\: \bf{10000 \; cm^{2}}}}}}$

$\: \\ \large{\boxed{\boxed{\sf{Total \; cost \; of \; Painting_{(Dome)} \;= \; \bf{Rate_{(in\; per\; m^{2})} \; \times\; Area\; to \; be \; painted}}}}}$

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★ Question :-

A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 22.8m, find the cost of painting it, given the cost of painting is Rs. 7 per 100 cm.

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★ Solution :-

Given,

» Circumference of the base of dome = 22.8 m

» Rate of painting the dome = Rs. 7 per 100 cm

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~ For the Radius of the Dome :-

• Let the radius of the base be 'r' m. Then,

$\: \\ \qquad \large{\sf{:\longrightarrow \: \;\: Circumference\; of \; Circle_{(base\; of \; Dome)} \: \; = \; \bf{2 \pi r}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \: \;\: 22.8 \; m\: \; = \; \bf{2\;\times\;\dfrac{22}{7}\;\times\;r}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \:\;\: r \;\: = \; \: \dfrac{\:22.8 \:m\;\times\; 7}{\:2\;\times\;22} \: \: = \: \: \bf{\underline{\underline{3.63 \; m}}}}}$

$\: \\ \large{\boxed{\sf{Radius\;\; of \;\; the \;\; dome\;\; = \; \bf{3.63 \; m}}}}$

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~ For the Area to be painted of the dome :-

$\: \\ \qquad \large{\sf{:\longrightarrow \;\:\; CSA \; of \; Hemisphere_{(Dome)} \:\; = \:\; \bf{2 \pi r^{2}}}}$

$\: \\ \qquad \large{\sf{:\longrightarrow \;\:\; CSA \; of \; Hemisphere_{(Dome)} \:\; = \:\; 2\;\times\;\dfrac{22}{7}\;\times\; (3.63\; m)^{2} \;\: = \;\: \underline{\underline{\bf{82.83\;m^{2}}}}}}$

$\: \\ \large{\boxed{\sf{Area\;\; to \;\; be \;\; painted_{(of \; dome)} \;\; = \; \bf{82.83 \; m^{2}}}}}$

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~ For the Actual rate of Painting :-

We are given the rate that is = Rs. 7 per 100 cm

We know that,

$\: \\ \sf{\rightarrow\;\;1 \; m \: = \:\; \bf{100\; cm}}$

So,

$\: \\ \sf{\rightarrow\;\;Rs.\: 7 \; per \; 100 \; cm \: = \:\; \bf{Rs.\: 7 \; per \; m}}$

Also, squaring both sides of this equation, we get,

$\: \\ \sf{\rightarrow\;\; (Rs.\: 7 \; per \; 100 \; cm)^{2}\: = \:\; \bf{(Rs.\: 7 \; per \; m)^{2}}}$

$\: \\ \sf{\rightarrow\;\; Rs.\: 49 \; per \; 10000 \; cm^{2}\: = \:\; \bf{Rs.\: 49 \; per \; m^{2}}}$

$\: \\ \large{\boxed{\sf{Actual\;\; rate \;\; of \;\; painting_{(in\:per\:m^{2})} \;\; = \; \bf{Rs.\; 49}}}}$

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~ For the Total cost of painting :-

$\: \\ \large{\sf{:\Longrightarrow \;\;\: Total \; cost \; of \; Painting_{(Dome)} \;= \; \bf{Rate_{(in\; per\; m^{2})} \; \times\; Area\; to \; be \; painted}}}$

$\: \\ \large{\sf{:\Longrightarrow \;\;\: Total \; cost \; of \; Painting_{(Dome)} \;= \; \bf{Rs.\: 49 \:\; \cancel{per\;m^{2}} \; \times\; 82.83\:\; \cancel{m^{2}} \:\; = \:\; \underline{\underline{Rs.\; 4058.67}}}}}$

$\: \\ \large{\underline{\underline{\rm{\mapsto \; \;\; Thus, \; total\; cost\; of \; painting\; is\;\; \boxed{\bf{Rs.\; 4058.67}}}}}}$

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$\: \large{\underbrace{\underbrace{\sf{Let's \; understand\; more \; formulas\; :-}}}}$

• Volume of Cone = ⅓ × πr²h

• Volume of Cylinder = πr²h

• Volume of Cuboid = Length × Breadth × Height

• Volume of Cube = (Side)³

• Volume of Sphere = (4/3) × πr³

Answer 2

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

• A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 22.8m, Find the cost of painting it, given the cost of painting is Rs. 7 per 100 cm.​

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

• Circumference of the base of the dome = 22.8m

• Cost of painting = Rs. 7 per 100 cm.​

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

• Cost of Painting hemispherical dome

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

• Cost of Painting hemispherical dome = Rs. 4058.67

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

• Let radius of dome be "r"

Circuference of Dome = Circuference of Circle with Radius "r"

⇒ 22.8 m = 2πr

⇒ 22.8 m = 2 × 22/7 × r  (∵ π = 22/7)

⇒ 22.8 m = 44/7 × r

⇒ 22.8 m × 7 = 44/7 × r × 7

⇒ 159.6 m = 44r

⇒ 159.6/44 m = 44r/44

⇒ 3.62727 m = r

⇒ r = 3.62727 m

⇒ r = 3.63 m (Rounding of to nearest hundredths)

Curved Surface of Dome = 2πr²

⇒ Curved Surface of Dome = 2 × 22/7 × r²

⇒ Curved Surface of Dome = 2 × 22/7 × (3.63 m)²

⇒ Curved Surface of Dome = 44/7 × (3.63 m)²

⇒ Curved Surface of Dome = [44 × (3.63 m)²]/7

⇒ Curved Surface of Dome = 579.7836 m²/7

⇒ Curved Surface of Dome = 82.82622 m²

⇒ Curved Surface of Dome = 82.83 m² (Rounding of to nearest hundredths)

We know :

• 1 m = 100 cm

Rs. 7 per 100 cm​ = Rs. 7 per 1 m​

⇒ Rs. 7 per 100 cm​ = Rs. 7 per  m​

⇒ (Rs. 7 per 100 cm)²​ = (Rs. 7 per m​)²

⇒ Rs. 49 per 10000 cm²​ = Rs. 49 per m​²

Cost of Painting hemispherical dome  = Rate in m​² × Curved Surface of Dome

⇒ Cost of Painting = Rs. 49 per m​² × 82.83 m²

⇒ Cost of Painting hemispherical dome = Rs. 4058.67.

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