Question

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

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

Radius of Greatest sphere = 7.20 meter

Step-by-step explanation:

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, ∠PMO= 90°

As, PR ⊥ QS, ∠PRS = 90°

In Δ PRS, and Δ PMO

m∠PRS = m∠PMO = 90°

∠MPR = ∠SPR   [Common Angles]

By AA postulate of Similarity of triangles. We get, ΔPRS ~ ΔPMO

As triangles are similar their sides will be proportional :

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

Hence, Radius of Greatest sphere = 7.20 meter