To calculate the cost of painting 50 hollow cones used for barricading a bus stop, first determine the slant height and surface area of one cone. With a base diameter of 40 cm (radius 20 cm) and height 1 m (100 cm), the slant height (l) using Pythagoras’ theorem is l = √[h² + r²] = √[100² + 20²] = √[10000 + 400] = √[10400] = 102 cm. The surface area of one cone is 3.14 × 20 × 102 = 6424 cm² or 0.6424 m². For 50 cones, the area is 50 × 0.6424 = 32.12 m². At ₹12 per m², the total cost is 32.12 × 12 = ₹385.44.

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## Introduction to the Project

The project involves painting 50 hollow cones made of recycled cardboard, which are used to barricade a bus stop. Each cone has specific dimensions: a base diameter of 40 cm and a height of 1 meter. The task is to calculate the total cost of painting the outer surface of these cones. The cost of painting is ₹12 per square meter. This scenario presents a practical application of geometric concepts in a real-world setting, combining the principles of surface area calculation with cost estimation.

### Understanding Cone Geometry

A cone is a three-dimensional geometric shape with a circular base and a pointed top, known as the vertex. The height of the cone is the perpendicular distance from the base to the vertex. The slant height, which is crucial for calculating the curved surface area, is the length of the line segment from the vertex to any point on the circumference of the base. In this case, the cones are hollow, implying that only the outer surface needs to be painted.

#### Calculating the Slant Height

The slant height of each cone is determined using Pythagoras’ theorem. For a cone with a base radius of 20 cm (half of the diameter) and a height of 100 cm, the slant height (l) can be calculated as l = √[h² + r²]. Substituting the values, we get l = √[100² + 20²] = √[10000 + 400] = √[10400] = 102 cm.

#### Determining the Surface Area of One Cone

The surface area of one cone is calculated using the formula for the curved surface area of a cone, πrl. With the radius of 20 cm and the slant height of 102 cm, the surface area for one cone is 3.14 × 20 × 102 = 6424 cm², which is equivalent to 0.6424 m². This area represents the part of the cone that needs to be painted.

##### Calculating the Total Surface Area for Painting

To find the total area that needs to be painted for all 50 cones, we multiply the surface area of one cone by the total number of cones. Therefore, the total area to be painted is
50 × 0.6424 = 32.12 m². This calculation is essential for determining the amount of paint required and the associated cost.

###### Calculating the Total Cost of Painting

The final step is to calculate the total cost of painting all the cones. With the painting cost set at ₹12 per square meter, the total cost for painting 32.12 m² is 32.12 × 12 = ₹385.44. This cost estimation is crucial for budgeting and resource allocation in the project. It demonstrates how geometric calculations are directly applied in practical scenarios, such as urban planning and public works, where efficiency and cost-effectiveness are key considerations.

Discuss this question in detail or visit to Class 9 Maths Chapter 11 for all questions.
Questions of 9th Maths Exercise 11.1 in Detail

 Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone. A conical tent is 10 m high and the radius of its base is 24 m. Find (i) slant height of the tent. (ii) Cost of the canvas required to make the tent, if the cost of 1 m² canvas is ₹70. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Use π = 3.14). The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of ₹210 per 100 m². A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is ₹12 per m², what will be the cost of painting all these cones?

Last Edited: January 2, 2024