Answer this : Three identical cones each have a radius of 50 and a height of 120 The - Maths - Conic Sections

Question

Answer this :

Three identical cones each have a radius of 50 and a height of 120
The cones are placed so that their circular bases are touching each
other A sphere is placed so that it rests in the space created by
the three cones, as shown If the top of the sphere is level with
the tops of the cones, then the radius of the sphere is closest
to

(A) 38 9 (B) 38 7 (C) 38 1 (D) 38 5 (E) 38 3

Mayank Jha · Accepted Answer

Dear student,

Let us represent the given information using a diagram
<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/ck-files/ck_57f5bda63329c%20jpg"
style="width: 390px; height: 532px;">

The centers of 3 circular bases form an equilateral triangle with side length 100
Let the circumcircle of this triangle have radius D
 Its radius is 100√3
O is the center of the sphere with radius r and Q is the point of contact of the sphere with one of the cones
The line PT is parallel to the base and forms a right triangle
AC has slope -125 so QO has slope 512OT = 5r13QT = 12r13The triangles ABC, APQ and QTO are similar
AP = S = S + OT = r + 5r13 = 18r13PQAP = 50120=512PQ = 512*AP = 512*18r13 = 15r26PT = PQ + QT = 15r26+12r13 = 3r2∵PT = BD
 So,3r2=100√3⇒r = 38
4900179 ≈38
5So, radius of the sphere is approximately 38
5 units

Regards