Question bank: Spatial Geometry: Surface Area of the Cone

An engineering company wants to build a conical silo for grain storage, aiming to optimize space and minimize the cost of the material used in the structure construction. The silo will have a height of 10 meters and the radius of the base will be variable, as the company intends to make silos of different sizes. Considering that the cost of the material used to build the lateral surface of a silo is proportional to the area of this surface and that the base area is not included in the cost, it is requested: 1) Develop an expression for the cost of the material as a function of the base radius 'r' and the height 'h' of the silo, knowing that the lateral area of a cone is given by 'A_cone = π*r*l', where 'l' is the generatrix of the cone. 2) Determine the base radius that minimizes the cost of the material needed to build the lateral surface of the silo.

A right circular cone has a base radius of 6 cm and its slant height is 10 cm. What is the total surface area of this cone, considering only 2 decimal places for the answer?

A packaging company has started producing paper cones for gift packaging. Each cone has a base radius of 5 cm and a height of 20 cm. To optimize the process of printing decorative patterns that will cover the entire outer surface of these cones, the company needs to calculate the total surface area of each cone. Based on the formula for calculating the surface area of a cone, which is A = πr(r + g), where 'A' is the surface area, 'π' is pi (approximately 3.14), 'r' is the base radius, and 'g' is the slant height of the cone, calculate the total surface area of a cone with the provided dimensions. Additionally, consider explaining how the area of a circle relates to the lateral surface area of the cone, and justify why we need to add the area of the base circle to the product 'πrg' to obtain the total surface area of the cone.

A technology company is planning to build a new data center with an innovative design that optimizes both space and cooling operations efficiency, and has decided that the data center will have the shape of a cone of revolution. The cone will have a height of 40 meters and the opening angle (in relation to the vertical axis) will be 75 degrees. Considering that the base of the cone will be made of a material that does not allow heat to pass through and that the cooling system efficiency depends on the cone's surface area, what will be the minimum lateral surface area of the cone that the project should specify to ensure efficient cooling of the data center? Consider π = 3.14 and round the result to the nearest hundred.

An architect is designing a new building and needs to calculate the amount of material needed to cover the surface of a water reservoir that will have the shape of a cone. The reservoir will be 10 meters high and the angle formed between the generatrix and the base of the cone is 60 degrees. To ensure waterproofing, he needs to determine the total surface area of the cone. Use the formula for the surface area of a cone, which is given by A = πr(r + l), where A is the surface area, r is the radius of the base, and l is the generatrix of the cone. To find the value of the radius, you can use the trigonometric relationship of the sine for the triangle formed by the generatrix, the radius of the base, and the provided angle. Consider π to be approximately 3.14 and do not forget that the generatrix of a cone is calculated through l = √(r^2 + h^2), where h is the height of the cone. After finding the surface area of the cone, calculate the amount of material in square meters that the architect will need to cover the reservoir.