The hemispherical dome of a building needs to be painted (see the figure). If the circumference of the base of the dome is 35.2 m, find the cost of painting it, given the cost of painting is ₹ 10 per 100 cm2.
Step-by-step solution
To find: Cost of painting the dome
Idea: Only the curved outer surface of the dome is painted: CSA = 2πr2. Get r from the circumference, and convert m2 to cm2 before using the rate.
- 2πr = 35.2 ⇒ r = 35.2 × 72 × 22 = 246.444 = 5.6 m.1 mark
- Area to paint = 2πr2 = 2 × 227 × 5.6 × 5.6 = 197.12 m2.1 mark
- 1 m2 = 10 000 cm2, so the area = 19,71,200 cm2.1 mark
- Cost = 19,71,200100 × ₹ 10 = 19 712 × ₹ 10 = ₹ 1,97,120.1 mark
Check: Rate ₹ 10 per 100 cm2 = ₹ 1000 per m2, and 197.12 × 1000 = ₹ 1,97,120 ✓.
Answer to write in the exam
2πr = 35.2 ⇒ r = 5.6 m
CSA = 2πr2 = 2 × 227 × 5.6 × 5.6 = 197.12 m2
= 197.12 × 10 000 = 19,71,200 cm2
Cost = 19,71,200100 × 10
∴ Cost = ₹ 1,97,120
Common mistakes that cost marks
- Using the total surface area 3πr2 (the flat base is not painted).
- Forgetting to change m2 to cm2 (1 m2 = 10 000 cm2, not 100 cm2).
- Taking 35.2 m as the diameter.
How this can come in the exam
The circumference of the base of a hemispherical dome is 44 m. Its curved surface area is (π = 227)
- 308 m2
- 616 m2
- 154 m2
- 924 m2
Show answer
(A) 308 m2
r = 7 m; 2 × 227 × 49 = 308 m2.
The inside of a hemispherical bowl of diameter 1.4 m is to be painted at ₹ 5 per 100 cm2. Find the cost. (π = 227)
Show answer
r = 0.7 m; 2πr2 = 2 × 227 × 0.49 = 3.08 m2 = 30 800 cm2. Cost = 308 × ₹ 5 = ₹ 1540.Try one yourself
A hemispherical dome has base circumference 17.6 m. Find the cost of painting it at ₹ 5 per 100 cm2. (π = 227)
Show answer
r = 2.8 m; CSA = 49.28 m2 = 4,92,800 cm2; cost = 4928 × ₹ 5 = ₹ 24,640.
More questions like this
- We learned about different 3D solids. Identify and list objects or structures around you that are cubical, cuboidal, cylindrical, conical, spherical, or pyramidal in shape, or closely resemble these shapes.
- Activity: Find the approximate volumes of a gas cylinder, a bucket, any cylindrical vessel, a cupboard, a water tank that you can find near you (use 1000 cm3 = 1 litre).
- Lallan’s golgappa (a small hollow ball made of wheat) stall has a cylindrical vessel with pani (spiced water) in it. The vessel is of radius 10 cm and it has pani up to a height of 50 cm. Estimate the number of golgappas he can serve assuming all golgappas are spheres. Before calculating, make a guess.
Can you guess what number this could be — is it in the hundreds, thousands, or more? - Anirban saw a few tennis balls lying around. He wondered, “How many tennis balls can fit in an empty classroom?”. The classroom is of dimensions 30 ft × 30 ft × 15 ft. Estimate how many tennis balls of radius 3 cm fit in this classroom. Before calculating, make a guess.
- Make an estimate based on the second and third configurations.