Question

The radius in cm of the greatest sphere that can be carved out of a solid cone of radius 9cm and height 40cm is

Answer 1

The Statement about the problem is

The greatest sphere that can be carved out of a solid cone of radius 9 cm and height 40 cm is:

Slant height of sphere = l²=h²+r²

l²= 40² +9²

l²=1681

l= 41 meter

Let the greatest sphere has radius R.

As sphere will touch the base as well as other two sides of the cone.

Line segment PS will be tangent to the sphere at point M.

So, ∠P MO= 90°

As, PR ⊥ Q S, ∠PRS= 90°

In Δ P RS, and Δ P MO

∠PRS=∠PMO=90°

∠MPR= ∠SPR →[Common Angles]

ΔPRS~ΔPMO[ AA Similarity]

As triangles are similar their sides will be proportional.

$\frac{PM}{PR}=\frac{MO}{RS}=\frac{PO}{PS}\\\\ \frac{R}{9}=\frac{40-R}{41}\\\\ 9(40-R)= 41 R\\\\ 50 R= 360 \\\\ R= 7.20$

Radius of Greatest sphere= 7.20 meter