Question

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A dome of a building is in the form of a hemisphere. From inside, it was white-washes at the cost of 4989.60. If the cost of white-washing is 20 per square metre, find the :

(i) inside surface area of the dome.
(ii) volume of the air inside the dome.


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Answer 1

Step-by-step explanation:

Question-:

A dome of a building is in the form of a hemisphere. From inside, it was white-washes at the cost of 4989.60. If the cost of white-washing is 20 per square metre, find the :

• (i) inside surface area of the dome.
• (ii) volume of the air inside the dome.

Solution✯

We know→

• → that Area white washed×Cost of white wash=Total cost

• →Area white washed×20=4989.60

• ☞Area white washed =204989.60=249.48‬‬m2

• ☞Now, area white washed is curved surface area of hemisphere(as only walls are whitewashed, not floor)

Now, area white washed is curved surface area of hemisphere(as only walls are whitewashed, not floor)∴→,inner surface area of dome =

$249.48‬‬m2$

→Volume of air inside dome=volume of hemisphere=

$32πr3$

☞Let the radius of dome be rm

→Let us first find the radius using surface area

Let us first find the radius using surface areaSurface area of dome=

$249.48‬‬m2$

⇒2πr2=249.48‬‬m2

⇒2πr2=249.48‬‬m2⇒2×722×r2=249.48‬‬m2

⇒2πr2=249.48‬‬m2⇒2×722×r2=249.48‬‬m2⇒r2=2×22249.48‬‬×7=39.69‬

⇒2πr2=249.48‬‬m2⇒2×722×r2=249.48‬‬m2⇒r2=2×22249.48‬‬×7=39.69‬⇒r=39.69‬=6.3m

• →Volume of the air inside the dome=Volume of hemisphere

Hence,

=32πr3

Required answer:- ✅

• =32×722×6.3‬×6.3×6.3m

$✯✯✯✯✯$

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Answer 2

Given:-

• The cost of white-washing the dome from inside = Rs. 4989.60
• The cost of white-washing 1 sq. metre = Rs. 20

To Find:-

1. inside surface area of the dome.
2. volume of the air inside the dome.

Formulas to be used:-

☆ Surface area of a hemisphere  = 2πr2

☆ Volume of a hemisphere  = 2/3πr3

Solution:-

1.) Inside surface area of the dome = Total cost for whitewashing the dome inside / cost of whitewashing 1 sq. metre

⇒ 4989.60/20

⇒ 249.48 m²

Hence, the inner surface area of the dome is 249.48m²

___________________________

2.) Let 'r' be the radius of the dome.

Inner surface area of the hemispherical dome = 2πr²

2πr² = 249.48 m²

⇒ r² =$\frac{249.48}{2\pi}$

⇒ r² =$\frac{249.48}{ \frac{2 \times 22}{7} }$

⇒ r² = $\frac{249.48 \times 7}{2 \times 22}$

⇒ r² = 39.69

⇒ r = √39.69

⇒ r = 6.3 m

The volume of the air inside the dome = the volume of the hemisphere.

The volume of the air inside the dome = 2/3πr³

= 2/3 × 22/7 × 6.3 m × 6.3 m × 6.3 m

= 523.9 m³ (approx.)

Therefore, the volume of the air inside the dome is 523.9 m³