#Mark the correct alternative in each of the following
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the he the cone to the diameter of the sphere is
In the figure, ∆PQR represents the 2D view of the cone and the circle represents the sphere. PA is perpendicular to QR and PS is diameter of circle. C will lie on PA due to symmetry.
Let the radius and height of cone be r and h and the radius of sphere be R. Also, the semi vertical angle of cone is α.
In ∆PAR
-(1)
∠APR=α
∠PAR=90°
Hence, ∠ PRA=180° -90° -α =90° -α
Also ∠PRS=90° (Angle in a semicircle)
Hence ∠ARS=∠PRS-∠PRA=α
In ∆RAS
AS=PS-PA=2R-h
–(2)
From (1) and (2), we get
r2=2Rh-h2
The volume of cone will be
Differentiating V with respect to h, we get
Differentiating V’ with respect to h, we get
For maxima at h=c, V’(c)=0 and v’’(c)<0
V’=0 ⇒
Hence, the ratio of height of cone to diameter of sphere is .