GATE ME 2017 Set 1 — Question 58
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
A right-angled cone (with base radius and height ), as shown in the figure below, is rolled on the ground keeping the point fixed until the point (at the base of the cone, as shown) touches the ground again.
By what angle (in radians) about does the cone travel?
Correct answer
(D) (10π)/(13)
Solution
The slant height of the cone is calculated using the Pythagorean theorem:
.When the cone rolls on the ground with its apex fixed, the base of the cone traces a circular path on the ground. The radius of this path is equal to the slant height of the cone, .The distance traveled by the base of the cone on the ground for one full rotation of the cone about its own axis (until point touches the ground again) is equal to the circumference of the cone's base:
.The angle (in radians) subtended at the apex by this arc length on a circle of radius is:
.
.When the cone rolls on the ground with its apex fixed, the base of the cone traces a circular path on the ground. The radius of this path is equal to the slant height of the cone, .The distance traveled by the base of the cone on the ground for one full rotation of the cone about its own axis (until point touches the ground again) is equal to the circumference of the cone's base:
.The angle (in radians) subtended at the apex by this arc length on a circle of radius is:
.
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