Sphere in cone

A sphere of radius 3 cm describes a cone with minimum volume. Determine the cone's dimensions.

Final Answer:

R =  4.24 cm
H =  12 cm

Step-by-step explanation:

r=3 cm tan Φ = R/H tan Φ/2 = 3/R  V = 31​ SH = 31​ π R2H V = 31​ R3 tan Φ V = 31​ (R/tan 2Φ​ )3 tan Φ  V′ = 9 π (tan 2Φ​)−3 ( cos2 Φ1​ − 23​ tan Φ arctan(Φ/2) cos2 (Φ/2)1​ ) V′ = 0  cos2( Φ/2 ) − 23​ tan Φ arctan(Φ/2) cos2 Φ = 0 cos ( Φ/2 ) ⋅  sin ( Φ/2 )  − 23​ cos Φ sin Φ = 0  21​ sin Φ − 23​ cos Φ sin Φ = 0 1−3 cos Φ = 0  Φ=arccos(31​)≐1.231 rad Φ = 70°31′44"  R=r/tan(Φ/2)=3/tan(1.231/2)=3 2​=4.24 cm
H=R⋅ tan(Φ)=4.2426⋅ tan1.231=12 cm

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You need to know the following knowledge to solve this word math problem:

geometryalgebrasolid geometryplanimetricsbasic operations and conceptsgoniometry and trigonometryUnits of physical quantitiesGrade of the word problem

 
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