Question bank: Spatial Geometry: Metric Relations of Cones

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 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 right cone has a volume of 100π cm³ and its height is twice the radius of the base. Considering that the relationship between the volume (V), the height (h), and the radius of the base (r) of a right cone is given by V = (1/3) π r²h, calculate the exact value of the height and the radius of the base of this cone. Then, check if the obtained measurements satisfy the condition that the height is twice the radius of the base.

Consider a right cone whose generatrix measures 10 cm and whose lateral surface is cut in such a way that, when opened and arranged on a plane, it forms a circular sector with a central angle of 60 degrees. Knowing that the line connecting the vertex of the cone to the midpoint of the base is perpendicular to the base plane and that the base radius measures 6 cm, determine: (1) The measure of the radius of the cone's cross-section formed by the cut. (2) The lateral surface area of the cone, considering π ≈ 3.14. (3) The total surface area of the cone, considering π ≈ 3.14.

A technology company specialized in cloud computing is designing cones for the assembly of antennas for a new satellite data transmission service. Each antenna, in a cylindrical shape with a base at the top, must be surrounded by a cone to direct the transmission at a specific angle towards the atmosphere. The critical dimensions of the cone are designated to ensure that the transmission beam is ideally directed, avoiding signal losses. The cone's opening angle, called 'solid angle,' is a measure of the space covered by the transmission beam in the atmosphere, and is calculated in terms of its aperture and height. Knowing that the relationship between the base radius and the height of a right cone is essential for calculating the solid angle, and that a measurement error in these dimensions could result in significant service degradation, determine the metric relationship between the base radius and the cone height that ensures the accuracy of the solid angle for the assembly of data transmission antennas. Also, consider that signal loss is unacceptable and must be minimized, and that data transmission is affected by atmospheric refraction, which takes into account the Earth's curvature and the variation of air density with height.