Question bank: Spatial Geometry: Metric Relations of the Cylinder

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Question 1:

Very Hard

An engineer is designing an irrigation system for a soccer field and decides that the shape of the system should be a cylinder, where the smaller base corresponds to the water inlet point and the larger base to the coverage area of the system. He needs to determine the amount of PVC pipes needed to connect all the water outlet points on the smaller base (one at each vertex of a regular polygon) to the inlet point on the larger base of the cylinder. The engineer knows that the smaller base has a regular polygon with 8 vertices and that the diagonal of the polygon measures 1 meter. In addition, the height of the cylinder is 5 meters. Using his knowledge of metric relationships in cylinders, calculate the distance that a PVC pipe must travel from the highest point of the smaller base to the inlet point on the larger base, justifying all steps of your reasoning.

Spatial Geometry: Metric Relations of the Cylinder

Theorical Question

Contextualized Question

Question 2:

Very Hard

A revolution cylinder has a generatrix of 10 cm and a base radius of 3 cm. An ant starts walking in a straight line on the bottom base of the cylinder at a distance of 1 cm from the farthest point from the axis of rotation. The ant walks on the surface of the cylinder, always maintaining the same distance from the axis, and stops at the diametrically opposite point on the upper base. What is the total distance traveled by the ant? Consider that the surface of the cylinder is smooth and the ant cannot fly or dig. Additionally, it always maintains a distance of 1 cm from the axis of rotation of the cylinder.

Spatial Geometry: Metric Relations of the Cylinder

Multidisciplinary Question

Question 3:

Medium

In a soda can factory, one of the engineers needs to calculate the distance between two opposite points on the bases of a cylinder formed by the can, to adjust the printing machine that will stamp the company's logo. Knowing that the can has a diameter of 6 cm and a height of 12 cm, what is the distance between two opposite points on the cylinder's bases?

Spatial Geometry: Metric Relations of the Cylinder

Question 4:

Very Hard

A right cylinder with a radius of 6 cm and a height of 10 cm is being used in a laboratory experiment. A straight line is drawn from a point A on the lower base of the cylinder to a point B on the upper base, such that the line forms a 30-degree angle with the height of the cylinder. Calculate the distance between points A and B, justifying all steps based on the cylinder's metric relationships and considering the sine and cosine of 30 degrees as precisely 1/2 and square root of 3/2, respectively.

Spatial Geometry: Metric Relations of the Cylinder

Contextualized Question

Calculus Question

Question 5:

Hard

A revolution cylinder is built from a rectangular sheet of cardboard, where one of the shorter sides of the sheet is glued to the opposite side of the same size, forming a base of the cylinder. The remaining ends of the rectangle are joined in such a way that the cylinder has a height equal to this shorter side. If the longer side of the rectangular sheet measures 24 cm and the shorter side measures 10 cm, calculate the distance between the farthest points of the two bases of the cylinder. Consider pi (π) as 3.14 and use the Pythagorean theorem to find the relationship between the base radius and the height of the cylinder.

Spatial Geometry: Metric Relations of the Cylinder

Contextualized Question

Calculus Question

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