Question bank: Spatial Geometry: Surface Area of the Cone

An ice cream factory is developing a new cone to serve its delicious creations. The cone is modeled by a conical surface, where it is observed that the height of the cone is twice the radius of the base. To ensure a uniform layer of ice cream, it is important to calculate the total area of this conical surface, including the area of the base. The radius of the cone's base is 5 cm. Determine the total area of the cone's surface and then indicate how the double of the base area relates to the lateral area of the cone.

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 developing a new cone-shaped server to optimize data storage. The cone has a base radius R and height H, and its base is closed. In order to decorate the external surfaces of the cone with a special material of high density and resistance, the company needs to calculate the total area of coating required. Given the formula for the lateral area of the cone as π * R * g, where g is the generatrix of the cone, and considering that the base area is π * R^2, determine the total coating area, in terms of R and H, that the company will need to acquire to coat 2 identical servers. Disregard the overlapping area between the coatings of the cone bases.

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 environmental engineer is designing a sprinkler irrigation system for a corn plantation in the shape of an inverted cone. The cone is 12m tall and the angle of the circular sector formed by the radius is 60º. To optimize the irrigation system, the engineer needs to know the total surface area of the cone, including the area of the base circle. The corn plant is circular, with a radius of 10m. Considering that the surface area of the cone is the sum of the areas of the base circle and the lateral surface, and that the formula for the lateral surface area of a cone is (pi * base radius * generatrix), where the generatrix is given by (square root of the square of the height plus the square of the base radius), calculate the total surface area of the cone so that the engineer can properly size the irrigation system. Consider pi as approximately 3.14 and use the formula for calculating the lateral surface area presented. Given the context presented, what would be the total surface area of the cone that the engineer should consider when designing the irrigation system for the inverted cone-shaped corn plantation?