Question bank: Spatial Geometry: Metric Relations of Cones

Consider a water reservoir in the shape of a right circular cone, and a field study that requires determining the volume of water stored in the reservoir. For this, it is assumed that the volume of a cone is given by the formula V = 1/3 π r^2 h, where V is the volume, r is the radius of the base, and h is the height of the cone. Knowing that the maximum storage capacity of the reservoir is 5000 liters and that the rate of change of the water level in the reservoir is 10 centimeters per hour, determine: (1) The rate of change of the water level in the reservoir in liters per hour. (2) If it is desired to increase the storage capacity of the reservoir to 6000 liters, by how many meters should the height of the cone be increased, keeping the same radius? Explain the steps of your reasoning and present a detailed calculation for each item.

A packaging company wants to create a new cardboard cone for storing food. They have specified that the volume of the cone should be exactly 1000 cm³ and that material economy suggests that the sum of the length of its generatrix with the height should be minimized. Based on this context, consider that the cone is formed from a circular sector of a circle with a radius of 10 cm and central angle α. (a) Determine the metric relationship that minimizes the sum of the generatrix with the height of the cone, relating α to these measures. (b) Calculate the minimum value of the sum of the generatrix with the height and discuss if this configuration is practical for the company in terms of manufacturing.

A municipality is planning the construction of a new cultural space, which will be a modern cone-shaped pavilion with a height of 30 meters and a base radius of 20 meters. At the top of the cone, a special lighting will be installed that needs to be visible from a minimum distance of 1000 meters in all directions. Considering that the luminosity follows a uniform pattern and that the height of the pavilion is the reference for the visibility calculation, and knowing that the lighting at the top of the cone creates an opening angle of 60º, determine the maximum height at which an observer can be located and still be able to see the top of the pavilion. Disregard the curvature of the Earth and the influence of obstacles in the field of view and assume that the observer and the top of the cone are located on the same flat surface tangent to the ground.

An ice cream container has the shape of a cone with a height of 15 cm and a base radius of 8 cm. A company wants to create a new design of an ice cream container that will maintain the same volume, but with a different base. They propose an ice cream container with a base in the shape of a circle with double the radius (16 cm) and its height being kept the same as the original container. To keep the volume constant, what should be the new height of the ice cream container with the new circular base?

Consider a right circular cone with height 'h' and base radius 'r'. When we cut the cone with a plane parallel to the base, we obtain a figure known as a 'frustum' or 'cone trunk'. Suppose we want to build a water reservoir in the shape of a cone trunk, where the area of the smaller base section is 'A' and the area of the larger base section is 'B', with 'A' < 'B'. The reservoir must have a capacity of 'C' cubic meters of water. (1) Prove that the height 'H' of the cone trunk can be expressed as a function of 'A', 'B', and 'h'. (2) Calculate the height 'H' in terms of 'A', 'B', 'h', and 'C' and determine the shapes that the cross-sections must have to ensure the desired amount of water, considering 'h', 'r', 'A', and 'B' as independent variables.