A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹500 per m².

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The design of a tent, which combines a cylindrical base with a conical top, presents a practical application of geometry. This problem involves calculating the area of the canvas needed to construct such a tent, a task that requires an understanding of the surface areas of both cylindrical and conical shapes. The challenge is not only to determine the amount of material required but also to calculate the cost of this material, providing insights into the practical aspects of geometric design in everyday objects. This exercise is a perfect example of how geometry is not just a theoretical subject but has real-world applications in various fields, including manufacturing and design.

The first step in our calculation is to analyze the cylindrical part of the tent. Given the height of 2.1 meters and a diameter of 4 meters, we can determine the radius as half the diameter, which is 2 meters. The surface area of the cylindrical part is calculated using the formula 2πrh, where r is the radius and h is the height. This formula calculates the lateral surface area of the cylinder, which is the part of the canvas that will wrap around the sides of the cylindrical base of the tent.

Next, we focus on the conical top of the tent. The slant height of the cone is given as 2.8 meters, and the radius is the same as that of the cylinder, 2 meters. The surface area of the cone is calculated using the formula πrl, where r is the radius and l is the slant height. This calculation gives us the area of the canvas required to cover the conical top of the tent. It’s important to note that this area represents the outer surface of the cone, which forms the top of the tent.

Total Surface Area of the Canvas
To find the total surface area of the canvas required for the tent, we add the surface areas of the cylindrical and conical parts. This involves summing the lateral surface area of the cylinder and the surface area of the cone. The total surface area represents the entire amount of canvas needed to construct the tent, excluding its base, as the base is typically left open.

After determining the total surface area of the canvas, the next step is to calculate the cost. The cost is computed by multiplying the total surface area by the rate of ₹500 per square meter. This calculation provides the total cost of the canvas needed to make the tent, offering a practical insight into the expenses involved in manufacturing a tent of this design. It’s a crucial step for budgeting and resource allocation in the production process.

In conclusion, the process of calculating the surface area and cost of the canvas for a tent is a clear demonstration of the practical application of geometric principles. This exercise not only reinforces our understanding of the surface areas of cylinders and cones but also highlights the importance of geometry in practical design and manufacturing. It showcases how theoretical geometric concepts are applied in real-world scenarios, emphasizing the role of mathematics in everyday life. This problem-solving approach is not just about numbers and formulas; it’s about understanding how geometry can be used to create functional and efficient designs.

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Questions of 10th Maths Exercise 12.1 in Detail