Competition problem — Geometry — Maths Olympiad Prep

Library / /1 of 94

Geometry Difficulty 3.7 AMC 10/12 Find the answer United States

Problem:

Find the volume of the three-dimensional solid given by the inequality $\sqrt{x^{2}+y^{2}} + |z| \leq 1$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

$2 \pi / 3$ . The solid consists of two cones, one whose base is the circle $x^{2}+y^{2}=1$ in the $xy$ -plane and whose vertex is $(0,0,1)$ , and the other with the same base but vertex $(0,0,-1)$ . Each cone has a base area of $\pi$ and a height of $1$ , for a volume of $\pi / 3$ , so the answer is $2 \pi / 3$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.