## Thursday, November 5, 2015

### Unizor - Geometry3D - Cones - Problems 1

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Right Circular Cones - Problems 1

In this course we will be dealing only with right circular cones and will call them simply cones.

Problem A

Find a volume of a truncated cone obtained by cutting-off its top by a plane parallel to its base in terms of two radiuses of two bases R1 and R2 and its height H (the distance between bases).

Answer:

πH(R1²+R1R2+R2²)/3

Problem B

The side surface of a cone, rolled out on a plane, is a circular sector with a central angle 120o and area S.

Determine the volume of this cone.

Answer:

2S√6πS /27π

Problem C

Given an area S of a side surface of a cone and a distance d from a center of its base to any generatrix, find a volume of a cone.

Answer:

S·d /3

Problem D

Side surface area of a cone is twice as big as an area of its base.

A section of a cone obtained by cutting it by a plane passing through its main axis (that is, a plane that goes through an apex and a center of a base) has an area S.

Find a volume of a cone.

Answer:

(27^1/4)πS√S/9

Problem E

An altitude of a cone of a volume V is divided into N equal parts by points Ai, where i∈[1,N−1]. For convenience, let A0 be an apex of a cone and AN - a center of its base.

Then we draw a plane through each division point parallel to a base of a cone. These planes divide the cone into N parts, all of them except the top are truncated cones.

Find the volume of a truncated cone #k between planes going through points Ak−1 and Ak.

Answer

V[k³−(k−1)³]/N³

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