To find the inside surface area and the volume of air inside a dome shaped like a hemisphere, we can use the given cost of white-washing and the rate per square meter.
(i) The cost of white-washing is ₹4989.60 at ₹20 per square meter. The surface area A can be found by dividing the total cost by the cost per square meter: A = 4989.60/20 m².
(ii) The volume V of a hemisphere is given by V = (2/3)πr³. To find the volume, we first need the radius, which can be derived from the surface area of a hemisphere A = 2πr². Solving for r and then substituting it into the volume formula will give the volume of air inside the dome. This approach demonstrates how geometric formulas can be applied to practical problems.

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## Geometric Problem Solving in Architecture

The application of geometry in architecture is evident in various structures, including domes. A dome, often hemispherical in shape, presents an interesting geometric problem when it comes to determining its surface area and internal volume. This is particularly relevant in tasks such as white-washing the interior of a dome, where understanding the surface area is crucial for cost estimation. Similarly, calculating the volume of air inside the dome can be important for ventilation and acoustics. Such problems demonstrate the practical application of geometric principles in architectural design and maintenance.

### Calculating the Surface Area for White-Washing

The cost of white-washing the interior of a dome provides a direct way to calculate its surface area. Given that the cost of white-washing is ₹4989.60 and the rate is ₹20 per square meter, the surface area can be determined by dividing the total cost by the rate per square meter. This calculation is based on the formula A = (Total Cost)/(Rate per Square Meter). It yields the total area that needs to be white-washed, a crucial piece of information for planning and budgeting the maintenance work.

#### Deriving the Radius from the Surface Area

The surface area of a hemisphere is given by A = 2πr², where r is the radius. By rearranging this formula, the radius can be calculated from the known surface area. This step is essential because the radius is a key variable needed to calculate the volume of the dome. Once the surface area is known from the cost calculation, the radius can be derived, providing a basis for further calculations related to the dome’s geometry.

Volume Calculation of the Dome
The volume of a hemisphere, representing the air inside the dome, is calculated using the formula V = (2/3)πr³. This formula highlights the relationship between the radius of the hemisphere and its volume. After determining the radius from the surface area calculation, it can be substituted into this volume formula. The resulting volume represents the total space inside the dome, which is an important factor in architectural design, especially for considerations like air circulation and acoustics.

##### Practical Implications in Architecture

Calculating the surface area and volume of a dome has practical implications in architecture and construction. The surface area is directly related to the cost of materials and labor for tasks like white-washing, while the volume is important for environmental control within the structure. These calculations are not just theoretical exercises but are essential for the practical aspects of building design, maintenance, and renovation. They enable architects and engineers to make informed decisions about material usage, cost estimation, and structural design.

###### The Intersection of Geometry and Architecture

In conclusion, the task of calculating the surface area for white-washing and the volume of air inside a dome exemplifies the intersection of geometry and architecture. These calculations are crucial for efficient planning, cost estimation, and ensuring the functionality of architectural structures. The application of geometric formulas to real-world problems in architecture demonstrates the importance of mathematics in practical scenarios. Understanding these concepts allows architects, engineers, and maintenance teams to effectively plan, design, and maintain architectural structures, highlighting the indispensable role of geometry in the built environment.

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Questions of 9th Maths Exercise 11.4 in Detail

 Find the volume of a sphere whose radius is (i) 7 cm (ii) 0.63 m. Find the amount of water displaced by a solid spherical ball of diameter (i) 28 cm (ii) 0.21 m. The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm³? The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon? How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold? A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank. Find the volume of a sphere whose surface area is 154 cm². A dome of a building is in the form of a hemisphere. From inside, it was white-washed 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. Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S′. Find the (i) radius r′ of the new sphere, (ii) ratio of S and S′. A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm³) is needed to fill this capsule?

Last Edited: January 5, 2024