Questão 136 - Caderno Azul | Geometria Espacial | MATEMÁTICA ENEM 2023

Summary

In this complex problem, an artist sculpts a sculpture from a block of wood in stages. Starting with a large cone, he removes a smaller cone and then a cylinder, leaving behind a truncated cone. The problem involves calculating the volume of the sculpture, considering the cone and cylinder dimensions, and using the density of the wood to determine the mass of the final sculpture. The process includes mathematical steps like volume calculations for cones and cylinders, and applying the given density to find the sculpture's mass. The final mass is determined to be 1296 grams.

Takeaways

• 😀 The artist carves a sculpture from a block of hardwood, starting with a large cone and progressively removing portions to shape it.
• 😀 The sculpture consists of a large cone, a smaller cone removed from the top, and a cylindrical section removed from the middle.
• 😀 The material used has a density of 0.6 g/cm³, and π is approximated as 3 for calculation purposes.
• 😀 The large cone has a height of 36 cm and a base diameter of 18 cm, leading to a radius of 9 cm.
• 😀 The smaller cone has a height of 12 cm and a base diameter of 6 cm, resulting in a radius of 3 cm.
• 😀 The large cone and the small cone are similar in shape, with the large cone being three times the size of the small one in terms of linear dimensions.
• 😀 The volume of the frustum (truncated cone) is calculated by subtracting the volume of the smaller cone from the volume of the larger cone.
• 😀 The volume of the cylinder removed from the middle of the frustum is calculated using the formula for the volume of a cylinder, with a radius of 3 cm and height of 24 cm.
• 😀 The total volume of the sculpture is found by subtracting the volume of the cylinder from the frustum's volume.
• 😀 The mass of the sculpture is determined by multiplying the final volume by the density of the material (0.6 g/cm³), resulting in a mass of 1296 grams.

Q & A

What is the shape of the initial object that the artist carves?

The initial object is a large cone with a height of 36 cm and a base diameter of 18 cm.

How does the artist create the truncated cone?

The artist removes a smaller cone from the top of the original large cone, leaving behind a truncated cone.

What is the diameter of the smaller cone that is removed?

The diameter of the smaller cone is 6 cm, and its radius is 3 cm.

What mathematical concept is used to calculate the volumes of the cones and the sculpture?

The volumes of the cones and the sculpture are calculated using the formula for the volume of a cone: V = (π * r² * h) / 3.

How is the volume of the truncated cone calculated?

The volume of the truncated cone is found by subtracting the volume of the smaller cone from the volume of the larger cone.

What is the significance of the ratio between the large cone and the small cone?

The large cone and the small cone are similar solids, with a linear ratio of 3:1. This ratio is used to determine the height of the small cone based on the height of the large cone.

How do you calculate the volume of the cylinder that is removed from the truncated cone?

The volume of the cylinder is calculated using the formula for the volume of a cylinder: V = π * r² * h, where the radius is 3 cm and the height is 24 cm.

What is the final volume of the sculpture after removing the cylinder?

The final volume of the sculpture is 2160 cm³ after subtracting the volume of the cylinder (648 cm³) from the volume of the truncated cone (2808 cm³).

What is the density of the wood used for the sculpture?

The density of the wood is 0.6 g/cm³.

How is the mass of the sculpture determined?

The mass of the sculpture is calculated by multiplying its volume (2160 cm³) by the density of the wood (0.6 g/cm³), resulting in a mass of 1296 grams.